IC-NRLF 


17    Elt 


UNIVERSITY  OF  CALIFORNIA. 


GIFT  OF 


U/*rxv/T 
^Accession 


Class 


AN  INQUIRY  INTO  THE  TEACHING 
OF  ADDITION  AND  SUBTRACTION 


THESIS 

Presented  to  the  Faculty  of  Philosophy  of  the  University  of 

Pennsylvania  in  Partial  Fulfillment  of  the 

Requirements  for  the  Degree  of 

Doctor  of  Philosophy 


ALBERT  DUNCAN  YOCUM 


PHILADELPHIA 
AVIL  PRINTING  COMPANY 

igoi 


AN  INQUIRY  INTO  THE  TEACHING 
OF  ADDITION  AND  SUBTRACTION 


THESIS 

Presented  to  the  Faculty  of  Philosophy  of  the  University  of 

Pennsylvania  in  Partial  Fulfillment  of  the 

Requirements  for  the  Degree  of 

Doctor  of  Philosophy 


ALBERT  DUNCAN  YOCUM 


PHILADELPHIA 
AVIL  PRINTING  COMPANY 

IQOI 


OF 

AVIL  PRINTING  COMPANY 

MARKET  AND  FORTIETH  STREETS 

PHILADELPHIA 


INTRODUCTORY. 

In  the  following  discussion  it  is  assumed  that  whatever 
the  numerical  knowledge  which  children  should  ultimately 
possess,  and  whatever  the  time  when  their  formal  instruc- 
tion in  number  should  begin,  they  must  at  the  very  outset 
become  thoroughly  familiar  with  the  fundamental  facts  of 
addition  and  subtraction ;  that  painstaking  inquiry  should 
be  made  into  the  numerical  content  of  their  minds  on  their 
first  entering  school;  that  so  far  as  possible  this  content 
should  become  the  basis  of  elementary  arithmetical  in- 
struction; that  aside  from  the  order  in  which  the  funda- 
mental number- facts  can  be  most  readily  mastered  by  each 
individual,  there  is  a  psychological  order  in  which  they 
should  be  taught  to  pupils  studying  en  masse — i.  e.,  that 
there  is  an  order  in  which  they  can  be  most  readily 
mastered  by  the  great  majority  of  individuals  who  are 
under  common  instruction;  that  similarly  there  is  a 
psychological  method  by  which  they  should  be  taught ;  and 
that  such  order  and  such  method  may  be  suggested  by  a 
priori  discussion  and  determined  by  practical  investiga- 
tion. 

This  inquiry  therefore  begins  with  a  resume  of  the 
results  of  all  investigations  into  the  numerical  content  of 
children's  minds  on  entering  school  which  were  accessible 
to  the  writer,  supplemented  by  the  data  obtained  in  a 
somewhat  restricted  investigation  of  his  own;  attempts 
to  prove  that  there  is  no  necessary  antagonism  between 
logical  order  and  psychological  order;  seeks  to  indicate 
what  on  a  priori  grounds,  should  be  the  psychological 
order  and  method  of  teaching  the  fundamental  facts  of 
addition  and  subtraction;  and  finally  reports  the  results 
of  the  use  of  the  order  and  method  thus  indicated,  in  the 
schools  of  a  large  manufacturing  town. 

(3) 


While  the  conclusions  resulting  from  this  inquiry  are 
purely  tentative,  and  may  of  course  be  disproved  by 
broader  investigation,  it  must  be  none  the  less  remem- 
bered that  they  are  based  upon  facts  and  have  been  suc- 
cessfully subjected  to  the  test  of  impartial  practice. 

Throughout  the  entire  discussion  effort  has  been  made 
to  avoid  all  difficulties  not  necessarily  involved  in  the 
simple  problem  which  it  seeks  to  solve.  It  cheerfully  leaves 
to  the  mathematical  philosopher  the  controversy  over  the 
number  'concept,  taking  it  for  granted,  however,  that 
such  concept  will  have  either  as  an  important  factor  or  a 
necessary  condition,  an  intelligent  knowledge  of  the  deci- 
mal system,  as  well  as  of  the  fact  that  each  number  in 
the  scale  is  greater  by  one  than  that  which  immediately 
precedes  it. 

With  equal  firmness  it  turns  aside  from  seductive  specu- 
lations concerning  the  origin  of  number  and  its  historic 
development,  assuming  that  even  should  the  Culture 
Epoch  Theory  be  regarded  as  demonstrated,  the  study  of 
the  child  would  be  more  likely  to  throw  light  upon  the 
history  of  number,  than  that  of  barbarous  dialects  and 
ancient  texts  upon  the  order  in  which  number  should  be 
taught  the  child. 

What  number  shall  be  taught  to  children,  whether  it 
shall  be  taught  them  on  their  first  entering  school,  what 
proportion  the  time  devoted  to  it  shall  bear  to  that  given  to 
other  branches  of  the  curriculum, — these  and  kindred 
questions  of  pedagogic  importance  it  fails  to  discuss,  not 
through  a  minimizing  of  their  importance,  but  from  the 
fact  that  they  are  judged  worthy  of  special  investigation. 

Certain  fundamental  sums  and  differences  must  be 
mastered  by  the  children.  Intelligently  derived,  they 
must  be  mechanically  used  in  numerical  operation.  If 
this  investigation  serves  to  indicate  an  order  and  a  method 
of  instruction  which  will  bring  about  their  readier  mastery 
by  the  majority  of  the  pupils  when  taught  in  common,  it 
will  accomplish  the  end  for  which  it  was  undertaken, — 


not,  it  is  hoped,  with  the  result  that  more  number-work 
will  be  done  in  a  given  school  year,  but  that  less  time  need 
be  devoted  to  the  number-work  at  present  demanded  by 
school  curricula,  whatever  that  time  may  be. 

It  is  only  proper  that  these  introductory  remarks  should 
conclude  with  formal  acknowledgment  of  the  practical 
assistance  afforded  by  Dr.  Lightner  Witmer,  Assistant 
Professor  of  Psychology,  and  the  suggestive  discussions 
which  characterize  his  seminaries.  To  him  and  to  Dr. 
Brumbaugh,  Professor  of  Pedagogy,  is  due  in  no  small 
measure  such  success  as  may  have  attended  the  inquiry. 


CHAPTER  I.— THE  NUMERICAL  CONTENT  OF 

CHILDREN'S  MINDS  UPON  THEIR  FIRST 

ENTERING  SCHOOL. 

From  1869,  when  the  Berlin  Pedagogical  Union  initi- 
ated the  first  general  inquiry  into  the  ideational  range 
(Vorstellungskreiz)  of  children  entering  school,1  such 
tests  have  been  common  in  the  pedagogic  centres  of  Ger- 
many. Nor  has  the  least  service  which  Dr.  Hall  has  done 
to  education  in  America  been  the  impetus  given  to  simi- 
lar investigation  through  the  forceful  demonstration  of  its 
value  and  suggestive  directions  as  to  its  method,  contained 
in  his  now  classical  report  on  the  tests  in  Boston  and 
Kansas  City  schools.2  Numerous,  however,  as  these 
tests  have  been,  they  have  shed  but  little  light  upon  the 
numerical  knowledge  possessed  by  children  at  the  begin- 
ning of  their  school  career.  Among  the  10,000  first 
tested  in  Berlin,  74.35  per  cent  were  found  to  be  familiar 
with  the  number  2 ;  73.99  per  cent,  with  3,  and  72.65  per 
cent,  with  4 — and  even  this  sparse  data  loses  much  of 
its  value,  when  it  is  known  that  some  of  the  returns  from 
which  it  was  obtained  were  made  after  the  children  had 
been  several  weeks  in  school.  These  are  the  only  facts 
concerning  children's  numerical  knowledge  which  the 
tests  sought  to  obtain,  and  yet  inquiry  was  made  into  their 
familiarity  with  almost  a  hundred  concepts.  When  one 
takes  into  account  the  proportionate  time  given  to  number- 
work  in  the  curriculum  of  the  elementary  school,  together 
with  the  fact  that  a  main  object  in  these  and  succeeding 
tests  was  to  determine  what  knowledge  the  beginner  could 
be  certainly  assumed  to  possess,  the  neglect  to  further  test 
his  numerical  knowledge  can  be  understood  only  by  assum- 
ing that  investigators  have  taken  it  for  granted — perhaps 
quite  unconsciously — that  the  child  enters  school  prac- 

(7) 


tically  ignorant  of  even  the  simplest  facts  of  number.  Dr. 
Lange  fails  to  touch  the  subject  of  number  at  all  in  his 
test  of  500  beginners  at  Flauen,  and  300  in  the  surround- 
ing country.3  Dr.  Hall  himself  after  a  searching  inquiry 
as  to  how  many  among  200  Boston  children  fail  to  have 
any  proper  concept  of  a  hundred  simple  and  familiar 
things,  only  adds  to  the  numerical  data,  resulting  from  the 
Berlin  test,  that  8  per  cent  of  his  subjects  did  not  know 
the  "number-name"  3;  17  per  cent,  4,  and  29.5  per 
cent,  5.4  This  he  does  because  "number  cannot  be 
developed  to  any  practical  extent  without  knowledge  of 
the  number-name,"5 — that  is,  he  assumes  a  minimum  of 
knowledge  necessary  to  the  teaching  of  number,  then 
seeks  to  discover  how  many  children  do  not  possess  it. 

Indeed  all  these  inquiries  into  the  general  content  of 
children's  minds  have  necessarily  been  negative  in  their 
results.  The  knowledge  which  children  may  possess  is 
so  extensive  and  so  various,  that  a  given  test,  even  though 
limited  to  that  part  of  the  possible  content,  likely  to  be 
common  to  a  majority  of  individuals,  while  showing  much 
that  the  individual  does  not  know,  can  show  but  little  of 
that  which  he  does.  On  the  contrary,  a  special  inquiry 
into  the  numerical  content  of  children's  minds  being  very 
limited  in  its  extent,  can  be  quite  positive  in  its  results. 

Another  characteristic  of  Dr.  Hall's  test  was  that  his 
200  children  were  carefully  selected,  being  limited  to 
those  of  "average  capacity,"  and  care  being  taken  to 
"avoid  schools  where  the  children  came  from  homes  repre- 
senting extremes  of  either  culture  or  ignorance."6  This 
usual  scientific  precaution  must  of  course  be  omitted  in  a 
special  inquiry,  which  seeks  among  other  things  to  deter- 
mine the  numerical  knowledge  which  can  be  counted  upon 
as  common  to  the  great  majority  of  beginners,  without 
previous  knowledge  as  to  their  capacity,  and  with  no 
regard  to  their  environment.  This  objection  does  not  hold 
against  the  results  of  Hartmann's  test  at  Annaberg  on  the 
660  boys  and  the  652  girls,  between  the  ages  of  six  and 


three-fourth  and  five  and  three-fourth  years,  who  began 
school  from  1881-84.  Of  these,  69  per  cent  of  the  boys 
and  62  per  cent  of  the  girls  could  count  from  i  to  io.7 
In  Superintendent  Greenwood's  tests  in  Kansas  City, 
identical  with  those  of  Dr.  Hall  and  reported  by  him,  the 
"number-names"  were  omitted, — probably  owing  to  the 
fact  that  at  the  time  of  the  test  the  children  had  been  some 
seven  months  in  school. 

It  would  seem  then  that  the  teacher, — discouraged  by 
the  difficulties  encountered  in  imparting  the  fundamental 
numerical  facts,  has  led  the  scientist  to  assume  that  the 
beginner  knows  little  or  nothing  of  number.  More  than 
this,  it  is  already  being  urged  on  what  may  be  or  may 
not  be  good  physiological  and  psychological  grounds,  that 
he  ought  to  know  nothing  of  it,  and  that  he  should  not  be 
taught  it  at  the  time  when  he  first  enters  school.  "The 
child,"  it  is  said,  "is  too  young  for  abstractions ;  he  is  still 
in  the  imaginative  epoch  of  his  cultural  development ;  cer- 
tain nerve  fibers  are  not  yet  medullated ;  nature  must  not 
be  forced," — an  argument  which  the  assumption  of  his 
numerical  ignorance  tends  somewhat  to  reinforce. 

But  since  aside  from  the  sparse  data  already  sum- 
marized, the  teacher  has  only  dogma  upon  which  to  de- 
pend, it  becomes  necessary  for  a  stable  pedagogy  to  deter- 
mine by  adequate  investigation  the  numerical  content  of 
children's  minds  on  their  first  entering  school. 

The  following  test  was  made  primarily  to  determine  in 
so  far  as  was  possible,  what  certain  individual  children 
knew  on  entering  the  schools  of  a  particular  locality,  and 
secondarily  as  a  step  in  the  more  general  determination  of 
what  numerical  knowledge  the  great  majority  of  Ameri- 
can children  may  be  assumed  to  have  in  common  on  enter- 
ing the  public  school.  For  obvious  reasons  it  was  not  made 
by  the  regular  teachers,  but  by  experienced  teachers' 
assistants,  who  had  been  carefully  instructed  together  in 
a  uniform  mode  of  procedure.  These  assistants  were 
seven  young  women — one  for  each  school, —  accustomed 


10 

to  the  management  of  children  and  selected  with  special 
reference  to  their  fitness  for  the  task.  The  results  obtained 
by  them  were  checked  by  the  regular  teachers  as  they 
began  the  work  of  the  year. 

The  test  was  made  on  the  first  day  of  school,  in  order 
that  its  results  might  be  reached  before  any  of  its  sub- 
jects had  been  in  charge  of  a  teacher,  or  had  been  given 
any  systematic  instruction.  Fifty  boys  and  fifty  girls 
were  subjected  to  it.  The  average  age  of  the  former 
was  5.64  years;  of  the  latter  5.48  years.  The  highest 
age  given  was  eight  years,  two  months.  Only  thirty 
children  were  older  than  six  years.  That  a  number  of 
children  admitted  were  below  school  age  there  can  be  little 
doubt,  as  many  hardworking  parents  look  upon  the  school 
as  a  day-nursery,  and  feel  that  a  misrepresentation  is 
excusable  which  will  relieve  them  from  serious  care,  and 
at  the  same  time  admit  their  children  a  little  earlier 
to  educational  advantages  which  they  must  relinquish  all 
too  soon.  The  test  was  made  in  a  manufacturing  town  of 
from  10,000  to  12,000  inhabitants,  comprising  an  un- 
usually small  proportion  of  professional  men,  but  by  way 
of  compensation,  very  few  foreigners.  The  children 
tested,  coming  from  every  section  of  the  town,  are  in  all 
probability  fairly  representative  of  the  great  mass  of 
native-born  beginners  in  the  country  as  a  whole. 

Great  care  was  taken  to  so  put  the  questions  that  the 
results  may  be  universally  valid.  Where  a  fact  was  put 
concretely  reference  was  strictly  limited  to  such  objects 
as  all  American  school  children  are  likely  to  have  counted 
— marbles  for  the  boys,  candies  and  pennies  for  both 
boys  and  girls.  A  possible  exception  is  to  be  found  in 
the  use  of  apple,  top,  and  orange  in  Questions  XII  to  XV. 
The  second  part  of  Question  XIX  is  theoretically  unfor- 
tunate owing  to  the  tendency  of  children  to  answer  "yes," 
when  they  think  that  an  affirmative  will  please.  Prac- 
tically, however,  this  tendency  would  be  hard  to  trace, — 
very  few  children  answering  the  question  at  all.  Ques- 


II 

tions  VII  and  VIII  should  have  been  supplemented  else- 
where by  others  in  which  the  facts  were  not  arranged  in 
series,  in  order  to  determine  how  far  regular  arrange- 
ment was  helpful  to  those  who  gave  them  correctly.  The 
later  experience  of  teachers,  however,  showed  conclusively 
to  what  extent  such  was  the  case.  As  a  possible  source  of 
serious  error  any  embarrassment  that  could  be  detected 
was  noted.  Most  children,  however,  were  natural  in 
their  manner,  and  seemed  to  respond  freely  to  questions 
which  they  were  able  to  answer. 

Suggestion  was  from  the  very  nature  of  the  questions 
impossible.  Prompting  was  permitted  in  counting,  but 
note  was  made  wherever  it  occurred.  No  conditions  of 
the  experiment,  therefore,  seem  such  as  to  seriously 
modify  its  results. 

As  the  recording  of  the  answers  given  was  so  planned  as 
to  involve  a  minimum  amount  of  writing,  each  investiga- 
tor filled  out  her  own  blanks.  Although  designed  as  they 
are  to  be  within  certain  limits  fairly  exhaustive,  the  ques- 
tions seem  quite  numerous,  it  will  be  seen  on  investiga- 
tion, that  it  was  unnecessary  to  ask  all  of  every  individual. 
The  average  time  taken  for  each  pupil  was  about  ten 
minutes.  The  following  is  a  copy  of  the  blank. 

MATHEMATICAL  CONTENT  OF  CHILDREN'S  MINDS  ON  THEIR 
FIRST  ENTERING  SCHOOL. 


I.  Age Years Months. . .... .... . . ...... 

II.  Boy  or  girl? 

III.  American  or  Foreign? 

IV.  Manner:   Natural  or  embarrassed?     (An  X  placed  above  a  word 

or  figure  will  indicate  it  as  the  answer;  above  a  fact  in  number, 
that  it  has  been  correctly  given;  above  a  question,  that  it  has 
been  correctly  answered.) 

V.  Can  he  count?    Yes  or  no.     To  what  number  without  objects? 

2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  14,  15,  16,  17,  18,  19,  20. 
(Cross  out  any  numbers  omitted  by  him,  or  on  which — while 
hesitating — he  was  prompted.) 


12 

VI.  To  what  number  can  he  count  actual  objects?     2,  3,  4,  5,  6,  7, 

8,  9,  10,  11,  12,  13,  14,  15,  16,  17,  18,  19,  20.  (Cross  out  as 
above.) 

VII.  Which  of  the  following  sums  can  he  give  correctly:    ("One  and 

one  more?"  "Two  and  one  more?"  etc.,  is  the  form  in  which 
these  sums  should  be  required;  first,  abstractly — correct  replies 
being  denoted  by  the  usual  X;  second,  concretely — correct  replies 
being  denoted  by  an  O  placed  after  the  X).  1  &  1,  2  &  1,  3  &  1, 
4  &  1,  5  &  1?  (Only  when  the  preceding  sums  are  correctly 
given  should  the  following  be  required) :  6&1,  7&1,  8&1, 
9&1,  10  &1,  11  &1,  12  &1,  13  &1,  14  &1,  15  &1,  16  &1, 
17  &  1,  18  &  1,  19  &  1,  20  &  1? 

VIII.  Which  of  the  following  differences  can  he  give  correctly?     ("If 
you  have  one  and  I  take  away  one,  how  many  have  you  left?" 
etc.,  is  the  form  in  which  these  differences  should  be  required, 
both  abstractly  and  concretely  as  above) .     1  —  1,2  —  1,3  —  1, 
4  —  1,  5  —  1,  6  —  1?     (Only  when  the  preceding  differences 
are  correctly  given  should  the  following   be  required) :   7  —  1, 
8  —  1,  9  —  1,  10  —  1,  11  —  1,  12  —  1,  13  —  1,  14  —  1,  15  —  1, 
16  —  1,  17  —  1,  18  —  1,  19  —  1,  20  —  1? 

IX.  Particular  Number  Associations. — Does  he  know  without  counting 

how  many  hands  he  has?     Yes,  or How  many  fingers 

on  one  hand?     Yes,  or The  number  of  legs  a  horse  has? 

Yes,  or (If  his  answer  to  one  of  the  questions  under  IX 

is  a  wrong  number,  put  it  in  the  blank;  if  so  to  one  under  X,  put 
it  above  each  successive  number  in  place  of  the  X.) 

X.  Number  Perception  too  Rapid  for  Conscious  Counting. — Can  he 

instantly  tell  the  number  of  splints  successively  shown  him  in 
groups  of  2,  4,  6,  3  and  5,  held  distinctly  apart  in  one  hand? 
The  total  number  when  2  are  thus  held  in  each  hand?  When 

3  are  so  held? 

XI.  Can  he  correctly  give  the  following  facts?      (First  abstractly,  then 

concretely,   and  indicated   as  above)  :     1  —  1,   2  —  2,   3  —  3, 

4  —  4,  5  —  5,  6  —  6?    2  &  2,  2  &  3,  3  &  2,  4  &  2,  2  &  4,  3  &  3? 
3  _  2,  4  —  2,  4  —  3,  5  —  2,  5  —  3,  5  — 4,  6  —  2,  6  —  4,  6  —  3, 
6  —  5? 

XII.  If  one  apple  costs  lc.,  what  will  two  apples  cost? 

XIII.  If  a  ball  costs  2c.,  what  will  three  balls  cost? 

XIV.  If  a  top  costs  2c.,  how  many  can  I  buy  for  4c.? 

XV.  If  an  orange  costs  3c.  how  many  can  I  buy  for  6c.? 

XVI.  2  ones?  2  twos?  2  threes?  3  ones?  3  twos? 


XVII.  How  many  twos  in  4?     In  6?     How  many  threes  in  6? 


XVIII.  You  have  three  sticks  and  I  have  two  sticks  —  Which  has  more? 
........  How  many  more?  ........    Which  less?  ........    How 

many  Ies3?  ........ 

XIX.  Into  how  many  parts  do  you  cut  an  apple  to  divide  it  in  half? 

........  Should  those  parts  be  of  the  same  size?  ........     To 

divide  it  into  thirds?  ........     Into  quarter??  ........ 


Name  of  Pupil .  .  . 
Name  of  Teacher. 


PERCENTAGES  FOR  FIFTY  BOYS  AND  FIFTY  GIRLS. 


V.  Can  Not  Count  at  All... 
Without 
With  Obi 

Can  Count  With  and  With 
Without  Objects 

VI.  With  Objects  From 

Can  Count  Higher  Withe 
With 

VII.  Know  Without  Objects. 

With  Objects 

VIII.  Know  Without  Objects 

With    Objects... 

IX.  Know  number  Hands .  .  . 
Fingers  .  . 
Horse's  Legs 


Boys. 
Per 
No.      Cent. 

4             8 
6         12 
2           4 
38        76 

Girls. 
Per 
No.      Cent. 

2            4 
10          20 
2           4 
36         72 

Objects  
ects 

out  Objects  .  . 

From  2  to  5  .  . 

39 

78 

38 

76 

5  to  10.. 

39 

78 

33 

66 

10  to  20.. 

31 

62 

26 

52 

2  to    5  .  . 

43 

86 

46 

92 

5  to  10.. 

37 

74 

36 

72 

10  to  20.. 

24 

48 

28 

56 

ut  Objects  .  .  . 

10 

20 

10 

20 

Objects  

16 

32 

19 

38 

....  1  +     .. 

14 

28 

19 

38 

2  +     .. 

20 

40 

19 

38 

to  10  +     .  . 

11 

22 

13 

26 

20  +     .. 

5 

10 

8 

16 

....   1  +     .. 

31 

62 

35 

70 

2  +     .. 

32 

64 

35 

70 

to  10  +  1  .. 

10 

20 

17 

34 

20  +  1  .  . 

4 

8 

7 

14 

....  2—    .. 

11 

22 

17 

34 

3—    .. 

11 

22 

14 

28 

to  10—    .. 
2—    .  . 

4 
18 
17 

8 
36 
34 

9 
25 
22 

18 
50 
44 

3  —    .. 

to  10  —  1.  . 

3 
41 
23 
39 

6 

82 
46 
78 

11 
41 
29 
38 

22 
82 
58 
76 

32S    . 

:    "    :                                                   Boys.  Girls. 

Per  Per 

No.  Cent.  No.  Cent. 

X.  Recognize  2 37  74  35  70 

3 27  54  30  60 

4 23  46  23  46 

5 11  22  11  22 

6 7  14  6  12 

2  Two  Groups 21  42  19  38 

2  Three  Groups 6  12  4  8 

XI.  Know  Without  Objects 1  —  1 . .     6  12  16  32 

2  —  2 . .     8  16  14  28 

to  6  —  6 . .     9  18  14  28 

With  Objects 1  —  1 . .   13  26  21  42 

2  — 2.  .   17  34  21  42 

to  6  — 6.  .   14  28  21  42 

Without  Objects 2  +  2..   12  24  6  12 

2  +  3..   12  24  5  10 

to3  +  3..     8  16  3  6 

With  Objects 2  +  2. .   10  20  14  28 

2  +  3..     6  12  18  36 

to  3  +  3..     5  10  6  12 

Without  Objects 3  —  2.  .   11  22  6  12 

4  —  2..     5  10  4  8 

to  6  —  5..     4  8  4  8 

With  Objects 3  —  2.  .   10  20  11  22 

4  — 2..     7  14  6  12 

to6  — 5..     4  8  6  12 

XII.  Know  1X2  Concretely 28  56  29  58 

XIII.  Know  2X3  Concretely 1  2  3  6 

XIV.  Know  2's  in  4  Concretely 4  8  7  14 

XV.  Know  3's  in  6  Concretely 3  6  3  6 

XVI.  Knowl  X  2  Abstractly 7  14  8  16 

2X2  Abstractly  4  8  3  6 

3X2  Abstractly 2  4  2  4 

1X3  Abstractly 3  6  3  6 

2X3  Abstractly 1  2  2  4 

XVII.  Know  2's  in  4  Abstractly 3  6  2  4 

2's  in  6  Abstractly 3  6  1  2 

3's  in  6  Abstractly 2  4  1  2 

XVIII.  Understand  Term  " More" 41  82  36  72 

"How  Many  More" 8  16  9  18 

Term  "Less" 14  28  11  22 

"How  Many  Less" 2436 


15 

Bays.  Girls. 

Per  Per 

No.       Cent.      No.     Cent. 

XIX.  Know  That  There  Are  Two  Halves  in 

Whole 27  54  18  36 

That  These  Parts  Should  Be 

Equal 18  36  11  22 

That  There  Are  Three  Thirds  in 

Whole  5  10  2  4 

That  There  Are  Four  Quarters 

in  Whole 5         10          2          4 

SUMMARY  OF  RESULTS. 

While  the  results  obtained  from  the  testing  of  so  small 
a  number  of  pupils  are  by  no  means  conclusive,  they  are 
none  the  less  significant.  No  generalizations  not  purely 
tentative  can  be  based  upon  them  alone ;  scientifically,  they 
are  merely  suggestive,  or  at  most  but  a  step  in  the  induc- 
tion of  positive  knowledge;  but  pedagogically  they  are 
final,  in  that  definitely  showing  the  numerical  knowledge 
of  certain  individual  children  who  are  to  be  taught 
together,  they  certainly  indicate  the  numerical  known  to 
which  the  unknown  facts  of  number  are  to  be  joined  and 
through  which  they  are  to  be  apperceived. 

The  first  significant  fact  to  note  is  that  the  girls  knew 
quite  as  much  of  number  as  the  boys, — if  not  somewhat 
more.  If  they  are  at  a  disadvantage  later  on  in  the  arith- 
metical course,  it  is  not  due  to  a  poor  start  as  compared 
with  that  of  their  brothers. 

Again,  as  was  to  be  expected, — there  was  noticeable  a 
great  variation  in  the  character  of  the  knowledge  dis- 
played by  various  individuals, — ranging  from  the  almost 
absolute  ignorance  of  the  three  or  four  who  knew  the 
least,  to  the  remarkable  readiness  with  which  two  or  three 
gave  the  most  difficult  facts.  In  the  case  of  the  majority, 
however,  the  knowledge  of  individual  facts  was  so  scat- 
tered and  unsystematic  as  to  compel  the  teachers  to  ignore 
it,  if  they  taught  in  the  usual  logical  order.  There  were 
few  children  who  did  not  have  a  considerable  knowledge 


i6 

of  number,  but  in  scarcely  any  two  cases  was  that  knowl- 
edge the  same.  Eighteen  per  cent  of  those  tested  knew  ab- 
stractly that  2  and  2  are  4,  and  17  per  cent  that  2  and  3  are 
5,  while  concretely  24  per  cent  knew  these  same  facts. 
Eleven  boys  and  six  girls  knew  3 — 2  as  an  abstract  fact, 
while  ten  boys  and  seven  girls  knew  it  in  the  concrete* 
But  all  those  who  knew  it  in  the  concrete  did  not 
know  it  in  the  abstract,  nor  did  the  same  boys  and  girls 
who  knew  the  differences  always  know  the  sums.  With 
any  method  of  teaching  which  took  up  in  regular  order  the 
combinations  and  separations  of  the  numbers  from  2  to 
20,  a  classification  which  would  from  the  start  group 
together  in  separate  divisions  pupils  having  approxi- 
mately the  same  degree  of  development  was  plainly  impos- 
sible. 

All  this  seems  at  first  glance  to  justify  the  assumption 
of  practical  ignorance  to  which  reference  has  already  been 
made,  and  the  consequent  practice — far  too  common — by 
which  with  few  exceptions  beginners  are  grouped  together 
at  the  start,  and  only  separated  into  divisions — if  sepa- 
rated at  all — when  they  are  found  to  widely  vary  in  the 
facility  with  which  they  acquire  the  elementary  facts. 

The  fact,  however,  none  the  less  remains,  that  the  ma- 
jority of  the  children,  thus  tested  on  entering  school,  dis- 
played a  considerable  knowledge  of  number.  There  were 
only  four  boys  and  two  girls  who  could  not  count  at  all, 
approximately  three-fourths  of  the  whole  number  could 
count  both  with  and  without  objects.  Of  those  who  could 
not,  six  boys  and  ten  girls  who  could  count  actual  objects, 
could  not  count  abstractly,  while  two  boys  and  two  girls 
who  could  count  abstractly,  could  not  count  actual  objects, 
— that  is  to  say,  with  but  4  per  cent  of  the  whole  number 
of  pupils,  was  the  counting  merely  mechanical.  Ten  boys 
and  ten  girls  could  count  higher  without  objects  than 
with;  sixteen  boys  and  nineteen  girls  higher  with  than 
without.  More  than  half  of  the  pupils,  however,  could 
count  equally  high  in  the  abstract  and  in  the  concrete.  As 


17 

full  comprehension  of  the  counting  process  can  call  for 
nothing  in  addition  to  this  but  the  generalization  that  each 
number  is  one  greater  than  that  immediately  below  it  in 
the  scale,13  it  is  surprising  to  learn  that  so  many  of  the 
children  possessed  the  knowledge  necessary  to  it.  While 
but  24  per  cent  of  the  whole  number  knew  that  2  cents  and 
2  cents  are  4  cents,  49  per  cent  knew  even  abstractly  that  2 
and  i  are  3,  while  62  per  cent  knew  that  2  cents  and  i  cent 
are  3  cents,  thus  beginning  a  generalization,  which  24  per 
cent  had  carried  to  10  and  i  abstractly  and  27  per  cent 
concretely,  and  respectively  13  per  cent  and  n  per  cent 
to  20  and  i.  Seventy-seven  per  cent  knew  that  2  is  more 
than  i,  and  17  per  cent  answered  correctly  what  must  have 
been  the  unfamiliar  question,  "How  many  more?"  It  was 
not  remarkable  that  but  25  per  cent  could  tell  whether  I 
was  "less"  than  2  or  2  than  i,  and  that  but  three  or  four 
knew  how  many  less.  Fifty- four  per  cent  of  the  boys  and 
36  per  cent  of  the  girls  knew  that  a  thing  to  be  divided  in 
half  must  be  cut  into  two  parts,  and  36  per  cent  of  the 
boys  and  22  per  cent  of  the  girls  understood  that  these 
parts  should  be  equal.  Ten  per  cent  of  the  boys  and  4 
per  cent  of  the  girls  knew  the  number  of  parts  into  which 
a  whole  must  be  cut  to  divide  it  into  thirds  or  into  fourths 
when  called  quarters. 

Although  1 8  per  cent  of  the  beginners  could  not  tell  how 
many  hands  they  had,  48  per  cent  the  number  of  fingers  on 
one  hand,  and  23  per  cent  the  number  of  a  horse's  legs, 
the  customary  assumption  that  such  lack  of  knowledge  is 
indicative  of  an  ignorance  of  number  was  not  proven.  On 
the  contrary  many  thus  ignorant  knew  as  much  as  those 
who  were  not,  their  ignorance  being  due  to  the  fact  that 
their  attention  had  not  been  directed  to  the  numbering 
of  the  particular  concrete  things  in  question.  Many 
adults  would  be  found  on  investigation  unable  to  give  the 
number  of  ribs  in  the  human  frame,  or  of  legs  normally 
possessed  by  a  fly. 

It  is  significant,  too,  that  72  per  cent  of  the  children 


i8 

recognized  the  number  2  with  counting  made  practically 
impossible;  57  per  cent,  the  number  3;  46  per  cent,  4; 
22  per  cent,  5 ;  and  13  per  cent,  6.  When  two  splints  were 
held  in  each  hand,  40  per  cent  recognized  the  total  as  4, 
and  when  three  were  so  held,  10  per  cent  the  total  as  6. 
With  but  very  few  exceptions,  all  children  who  knew  the 
number  of  hands  they  had,  could  recognize  2  without 
counting,  and  all  who  knew  a  horse  had  four  legs  could 
so  recognize  4 — something  which  scarce  a  child  without 
that  knowledge  could  do.  It  is  significant  in  this  connec- 
tion that  while  77  per  cent  knew  the  number  of  legs  a 
horse  has,  though  horses'  legs  are  in  pairs,  only  24  per 
cent  knew  that  2  and  2  are  4,  and  7  per  cent,  two  2's. 

Incidentally  acquired  though  this  knowledge  is,  scant 
in  the  case  of  the  few,  but  quite  considerable  in  that  of  the 
majority,  if  repeated  tests  in  various  localities  confirm  the 
foregoing  results,  it  will  at  least  be  certainly  demonstrated 
that  the  great  majority  of  children  know  something  of 
number  on  entering  school.  Whether  they  have  enough 
in  common  for  the  public  school  teacher  to  be  able  to 
utilize  it,  will  be  discussed  later. 

COMMENTS  UPON  THE  FOREGOING  RESULTS. 

Whatever  the  facts  that  may  be  ultimately  associated 
with  a  given  digit,  however  various  the  numerical  relations 
in  which  it  plays  a  part,  primarily  it  seems  to  stand  for 
aggregation.  The  child  knows  that  i  and  i  are  2  before 
he  learns  that  two  I's  are  2 ;  that  5  and  i  are  6  before  he 
discovers  or  is  led  to  discover  that  three  2's  are  6.  He 
asks  for  "another"  or  "one  more"  rather  than  for  two 
times  as  many  as  he  had  before.  He  learns  to  count  before 
he  masters  the  elementary  products.  Multiplication  as 
taught  him  in  the  school  may  never  be  to  him  "a  short 
method  of  addition,"  but  nevertheless  were  he  left  to  him- 
self, 8  would  be  4  and  4,  before  it  would  be  4  taken  two 
times,  or  multiplied  by  2.  It  has  been  left  for  the  philos- 


19 

opher  to  assert  the  fact  that  number  is  ratio.  The  child 
counts  before  he  measures.  To  him  addition  is  counting 
on,  subtraction  is  counting  off.  Counting,  then,  is  the 
fundamental  process.8  It  begins — in  so  far  as  it  is  not 
merely  mechanical — with  the  child's  plea  for  "just  one 
more/'  his  perception  of  one  thing  and  another  one,  not,  of 
course,  with  the  abstraction  I  and  i.9  Sixty-six  per 
cent  of  the  100  children  tested  on  entering  school  knew  that 
i  cent  and  another  cent  are  2  cents ;  only  33  per  cent  that 
i  and  i  are  2.  Half,  however,  of  those  who  knew  this 
primal  fact  of  number  at  all,  had  made  the  generaliza- 
tion necessary  to  apply  it  to  any  one  thing  and  another, — 
a  fact  which  is  of  significance  in  a  later  connection.  While 
only  four  boys  and  two  girls  were  unable  to  count  at  all, 
thirty-four  boys  and  girls  did  not  know  that  i  more  than 
i  familiar  thing,  or  i  familiar  object  and  another, — are 
2.  Hence  it  is  plainly  obvious  that  many  children  who 
are  reported  as  being  able  to  count  were  completely 
ignorant  of  the  fact  that  each  successive  digit  is  one  more 
than  the  one  which  immediately  precedes  it.  It  would  be 
erroneous,  however,  to  assume  that  counting  thus  unintel- 
ligent, is  a  purely  mechanical  repetition.  Only  16  per 
cent  of  the  whole  number  of  children  were  unable  to  count 
actual  objects,  and  thus  to  definitely  number  a  group  of 
I's,  of  whose  number  as  compared  with  other  groups  they 
had  no  knowledge.  For  example,  the  great  majority  of 
beginners  were  at  least  able  to  determine  that  there  were 
four  objects  in  a  given  group,  though  ignorant  oi  the  fact 
that  their  number  was  i  more  than  that  of  a  group  of  3 
or  i  less  than  that  of  a  group  of  5,  while  a  very  consider- 
able number  had  made  the  generalization  necessary  to 
intelligent  counting.  These  results  tend  to  demonstrate 
that  upon  entering  school,  a  few  children  will  be  found 
who  cannot  count  at  all  (in  this  case  6  per  cent)  ;10  a  few 
whose  counting  is  merely  mechanical  repetition  originat- 
ing in  the  series  concept,8  conventionalized  by  imitation, 
and  stimulated  by  a  love  of  rhythm  (  16  per  cent)  j11  a 


2O 


large  number  who  having  taken  the  next  step  can  by 
counting  determine  within  narrow  limits  the  number  of 
objects  in  a  given  group  though  ignorant  that  each  digit 
which  they  name  is  greater  by  I  than  that  immediately 
preceding  it  (34  per  cent)  ;12  a  still  larger  number  (44 
per  cent),  comprising  almost  half  of  the  children  tested, 
— who  have  at  least  begun  to  count  intelligently.13 

Counting,  then,  it  would  seem,  is  at  first  to  the  great 
majority  of  children  a  rhythmical  series  of  names  with 
little  or  no  meaning,  sooner  or  later  correctly  applied  to 
particular  groups  of  objects,  and  gradually  perceived  as 
a  series  of  number-names  each  one  of  which  may  be 
applied  to  a  group  containing  one  more  object  than  does 
that  to  which  the  number-name  last  preceding  it  maybe  ap- 
plied. To  some,  counting  may  never  be  a  mechanical  pro- 
cess. They  may  successively  add  by  I's,  may  know  that  i 
and  i  are  2,  2  and  i,  3,  etc.,  before  they  repeat  i,  2,  3,  as  a 
series.  Indeed  this  addition  of  successive  I's  and  the  me- 
chanical counting  may  go  on  side  by  side,  without  the 
child's  perceiving  any  connection  between  them. 

The  fact  that  "counting  on"  or  "forwards"  is  so  popular 
a  process  with  children  is  likely  to  make  it  a  matter  of 
imitation  before  it  becomes  intelligent;  not  so,  however, 
with  "counting  off."  Very  few  of  the  children  tested 
could  count  backwards  from  10  to  i,  yet  28  per  cent  knew 
in  the  abstract  that  2  less  i  are  i,  25  per  cent  that  3  less 
i  are  2,  and  13  per  cent  the  most  of  the  remaining  facts  to 
10  less  i,  while  in  the  concrete  the  percentages  were 
severally  43  per  cent,  39  per  cent  and  14  per  cent.  The 
fact  that  few  children  knew  these  differences  in  any 
systematic  order  strongly  indicates  that  they  were 
mastered  as  separate  and  distinct  facts.  They  did  not 
seem  to  know  that  3  less  i  are  2,  because  2  is  next  below  3 
in  the  digit  series.  Unlike  the  sums  formed  by  adding  i, 
there  was  nothing  so  far  as  could  be  seen — to  differentiate 
the  facts  in  which  i  was  subtrahend  from  those  in  which 
the  subtrahend  was  2  or  3.  It  is  evident,  then,  that  the 


21 


only  way  in  which  these  children  became  familiar  with 
certain  differences  was  the  process  by  which  they  came  to 
know  such  of  the  elementary  sums  as  were  not  obtained  by 
adding  i.  Just  as  they  counted  out  the  number  of  objects 
to  be  added  and  then  counted  out  their  sum,  so  they  must 
have  counted  out  the  number  of  objects  in  the  minuend, 
the  number  that  was  taken  away  and  the  number  remain- 
ing. They  were  almost  without  exception,  ignorant  of 
the  term  "less ;"  subtraction  was  to  them  a  "taking  away" 
with  a  certain  remainder,  not  the  lessening  of  a  number  by 
a  certain  amount.  The  number  remaining  does  not  seem 
to  have  been  inferred  from  the  number  to  which  the  sub- 
trahend is  added  to  obtain  the  minuend.  That  is,  they  did 
not  know  that  3  less  2  is  i ,  because  they  already  knew  that 
2  and  i  are  3,  but  because  they  met  with  it  in  their  experi- 
ence as  a  separate  and  distinct  fact.  Twenty-three  per 
cent  abstractly  and  35  per  cent  concretely  seem  to  have 
made  the  generalization  that  when  a  number  is  taken  from 
itself  the  remainder  is  "nothing." 

However  few  the  disconnected  facts  of  number  which 
a  child  masters  as  distinct  operations,  and  which  repeated 
in  his  experience  with  different  objects  become  abstrac- 
tions, it  is  probable  that  they  would  be  fewer  still  did  he 
have  to  laboriously  count  out  every  number  which  they 
involve.  Three-fourths  of  the  children  tested  could  recog- 
nize 2  without  noticeable  counting;  more  than  half,  3; 
and  a  little  less  than  half,  4.  Twenty-two  per  cent  and 
13  per  cent  respectively  thus  recognized  5  and  6.  As  the 
objects  used  in  this  test  were  splints  held  fan-like  and  dis- 
played but  for  an  instant,  the  resulting  forms  were  hardly 
those  that  the  child  would  already  have  associated  with 
particular  number-names.  This  being  so,  while  I  fully 
agree  with  Dr.  Phillips  that  much  which  passes  as  number 
recognition  "is  only  the  recognition  of  an  individual 
form/'14  in  the  present  case  the  children  seem  to  have 
recognized  number.  It  may  be  more  than  a  mere  coinci- 
dence, however,  that  with  but  very  few  exceptions,  all 


22 


children  who  knew  the  number  of  hands  they  had,  could 
recognize  2,  and  that  all  who  knew  a  horse  has  four  legs, 
could  recognize  4, — something  which  scarce  a  child  with- 
out that  knowledge  could  do.  Two  objects  can  not  be 
so  placed  that  they  will  not  be  in  a  straight  line,  three  and 
four  objects  are  usually  so  grouped  that  they  may  respec- 
tively form  vertices  for  triangles  and  quadrilaterals.  The 
relative  position  of  the  horse's  legs,  human  hands  or  the 
feet  of  a  three-legged  stool  is  not  different  from  that  of  the 
corresponding  number  of  splints. 

In  the  absence  of  recognition  by  the  children  of  what  is 
thus  in  effect  an  imaginary  general  form, — far-fetched 
though  its  assumption  may  seem, — whether  we  grant  the 
truth  of  Kulpe's  hypothesis  of  a  direct  recognition,  de- 
pendent upon  the  "effectiveness"  of  a  percept  for  central 
excitation  and  the  "mood"  which  it  induces,15  or  assume 
that  the  recognition  is  here  due  to  the  comparison  of  the 
present  percept  with  a  memory  image  in  part  identical, — 
an  adequate  explanation  of  this  instantaneous  number 
recognition  is  exceedingly  difficult.  In  the  first  place, 
whatever  may  be  true  in  the  case  of  the  adult,  in  that  of 
the  child  just  entering  school,  there  is  little  likelihood  of 
unconscious  counting,  and  no  possibility  whatever  of 
either  conscious  or  unconscious  estimation.  Few  of  the 
children  tested  were  orally  rapid  counters,  and  the  great 
majority  in  counting  objects  assisted  discrimination  by 
pointing  to  each  successive  one  or  by  handling  it.  It  is 
altogether  unlikely  that  a  "mental  counting"  of  which 
there  was  no  external  sign — would  be  rapid,  if  indeed  in 
the  case  of  the  majority — it  could  even  be  possible  with 
any  degree  of  accuracy. 

Tests  in  which  the  pupils  would  be  led  to  so  count  the 
numbers  certainly  recognized  would  be  indeterminate,  in 
that  in  the  absence  of  introspection  the  observer  would 
not  determine  whether  the  result  was  reached  through 
counting  or  not,  while  if  they  were  found  able  to 
"mentally"  count  higher  numbers,  the  results  would  not 


23 

necessarily  apply  to  the  lesser  numbers  directly  recog- 
nized. 

Unconscious  counting  being  thus  improbable,  uncon- 
scious estimation  was  plainly  impossible,  where  a  ma- 
jority of  the  subjects  were  ignorant  of  the  necessary  facts. 
The  grouping  of  objects  instead  of  helping  to  a  more 
accurate  judgment,  resulted  in  the  reverse, — only  40  per 
cent  of  the  pupils  being  able  to  recognize  two  2  groups 
as  4,  and  10  per  cent  two  3  groups  as  6.  The  separation 
into  two  groups  only  made  it  more  difficult  to  perceive  the 
objects  as  one  whole. 

While  it  may  be  well  to  recall  in  this  connection  the 
famous  experiments  in  which  young  children  through 
the  display  of  a  number  of  large  and  irregular  objects  side 
by  side  with  a  larger  number  of  smaller  objects,  have  been 
deceived  into  pronouncing  the  former  number  the  greater, 
they  merely  serve  to  demonstrate  that  children,  like  adults, 
may  be  led  to  form  a  wrong  judgment  of  comparative 
number. 

In  conclusion,  then,  we  may  state  as  true  of  the  particu- 
lar children  tested  and  tentatively  assume  as  true  of 
American  children  in  general: 

1.  That  the  majority  of  children  have  some  knowledge 
of  number  on  first  entering  school. 

2.  That  to  them  counting  is  the  initial  numerical  pro- 
cess, and  the  ability  to  count,  the  only  numerical  knowl- 
edge which  they  have  in  common. 

3.  That  they  have  little  or  no  knowledge  of  ratio. 

4.  That  to  them  the  numbers  i,  2,  3  and  4  have  become 
concepts  which  in  most  cases  can  be  instantaneously  and 
correctly  applied  whenever  the  corresponding  groups  of 
objects  are  distinctly  seen. 


CHAPTER  II.— AN  A  PRIORI  DETERMINATION 

OF  THE  PSYCHOLOGICAL  ORDER  OF 

TEACHING  NUMBER. 

I.  DISCUSSION   OF  LOGICAL  AND   OF  PSYCHOLOGICAL 

ORDER. 

As  has  been  already  asserted,  it  is  impossible  to  utilize 
the  numerical  knowledge  possessed  by  children  on  enter- 
ing school,  if  number  is  to  be  taught  them  in  the  usual 
logical  order, — that  is,  in  the  order  which  exhausts  the 
combinations  and  separations  of  each  successive  number 
from  i  to  20  before  the  teaching  of  any  fact  connected 
with  those  above  it.  The  question  therefore  naturally 
arises,  is  this  logical  order  the  only  order  in  which  number 
can  be  taught,  and  if  not,  is  it — all  things  considered — 
the  order  in  which  the  fundamental  facts  of  addition  and 
subtraction  can  be  most  readily  mastered  by  beginners? 
There  is  a  tendency  among  disciples  of  "the  new  educa- 
tion" to  take  for  granted,  an  antagonism  between  "logical" 
and  "psychological"  order.  The  knowledge  of  the 
primary  school,  it  is  assumed,  must  not  be  systematic 
knowledge,  because  the  order  in  which  facts  are  best 
grouped  by  the  adult  mind  is  not  that  in  which  they  can 
be  most  readily  apperceived  by  the  mind  of  the  child. 
Now  it  has  by  no  means  been  demonstrated  that  knowl- 
edge logically  arranged  can  not  be  more  readily  compre- 
hended by  the  beginner  than  can  any  aggregation  of 
isolated  facts,  however  simple  those  facts  may  be.  It  is 
characteristic  of  a  logical  order  that  facts  having  some- 
thing in  common  are  grouped  together.  A  logical  classi- 
fication, basing  itself  on  an  element  common  to  a  large 

(24) 


25 

number  of  things,  will  separate  out  from  the  mass  two  or 
more  groups  which  will  together  contain  all  things  having 
the  common  element,  but  which  are  differentiated  from 
each  other  in  that  the  things  in  each  group  have  an  addi- 
tional common  element,  which  those  in  the  others  have 
not.  Similarly  each  group  may  then  be  separated  into 
additional  groups,  the  partial  identity  of  the  things  in  each 
sub-group  growing  greater  and  their  number  less,  until  it 
is  impossible  for  partial  identity  to  be  thus  increased. 
Facts  are  taught  in  a  logical  order,  when  beginning  with 
the  things  having  least  in  common,  instruction  succes- 
sively passes  to  those  which  have  most,  or  when  beginning 
with  those  which  have  the  most  in  common,  it  similarly 
passes  to  those  which  have  least.  Teaching  which  thus 
goes  from  the  general  to  the  particular  is  called  deductive ; 
that  which  taking  the  opposite  course,  passes  from  the 
particular  to  the  general,  is  known  as  inductive.  As  the 
things  included  in  a  given  class  may  be  differently  grouped 
accordingly  as  this  or  that  element  common  to  many  of 
them,  is  made  the  basis  of  classification,  the  same  things 
may  usually  be  classified  in  many  different  ways,  with  the 
result  that  the  young  may  be  taught  both  deductively  and 
inductively  in  more  than  one  logical  order.  Now  aside 
from  such  general  conditions  as  attention  and  fatigue, 
nothing  is  more  necessary  to  the  processes  involved  in 
apperception  and  recall,  than  such  arrangement  of  indi- 
vidual facts  as  makes  more  readily  perceivable  the  presence 
of  some  common  element. 

It  is  only  when  the  common  element  is  too  insignificant 
for  children  to  readily  perceive  it,  or  when  the  things  to 
which  it  is  common  are  too  complex  or  too  various  for  it 
to  be  readily  perceived,  that  facts  logically  grouped  can  not 
be  readily  mastered.  The  psychological  order  of  teaching 
a  number  of  particular  facts,  is  that  through  which  a  given 
individual  or  group  of  individuals  can  most  readily  ap- 
proximate their  full  apperception  and  prompt  recall. 
While  the  psychological  order  thus  concerns  itself  with 


v 

OF  THE 


26 

the  succession  in  which  facts  are  presented  to  the  indi- 
vidual mind,  and  the  logical  with  the  manner  in  which 
they  are  related  to  each  other,  it  is  readily  to  be  seen  that 
the  former  is  conditioned  by  the  latter. 

Not  only,  therefore,  is  there  no  necessary  antagonism 
between  the  logical  and  the  psychological  orders,  but  it 
may  be  seriously  questioned  whether  to  the  vast  majority 
of  individuals  any  order  of  teaching  the  facts  of  the 
school  curriculum  is  fully  psychological  without  being 
logical.  If  the  facts  to  be  imparted  to  a  given  individual 
have  no  element  in  common,  that  order  of  teaching  will  be 
psychological,  which  so  presents  them  that  each  separate 
fact  will  precede  all  others  more  difficult  for  that  individ- 
ual to  perceive.  If  on  the  contrary  the  facts  to  be  thus  im- 
parted have  anything  in  common,  which  mastered  in  one 
will  materially  aid  in  the  mastery  of  all,  that  order  of 
teaching  will  be  psychological,  which  groups  together  the 
facts  possessing  this  common  element,  provided  that  it  is 
readily  perceivable  to  the  individual  taught.  That  such 
a  grouping  is  logical  is  evident  per  se.  Since  the  great 
majority  of  facts  included  within  the  school  curriculum, 
have  more  or  less  in  common  with  other  facts,  it  neces- 
sarily follows  that  wherever  possible,  they  must  be  pre- 
sented in  a  logical  order  whose  common  element  readily 
perceived  by  the  majority  of  pupils,  in  the  mastery  of  one 
or  more  individual  facts,  will  insure  the  ready  mastery  of 
the  remaining  facts  with  which  they  are  grouped.  It  will 
be  convenient  to  designate  such  a  common  element  as  an 
"effective"  element. 

The  logical  order  will  therefore  be  psychological,  when 
its  common  element  is  one  thus  effective,  and  readily  per- 
ceivable to  the  individuals  taught.  Other  things  being 
equal,  the  larger  the  number  of  the  facts  included  in  its 
several  groups,  the  more  psychological  will  an  order  be, 
for  the  larger  the  number  of  facts  in  the  groups,  the 
smaller  the  number  of  groups,  and  hence  the  readier  the 
mastery  of  the  whole  body  of  facts.  Moreover  it  is  plain 


27 

that  the  common  element  should  not  only  be  effective,  but 
that  it  should  be  that  particular  effective  element  which  is 
the  basis  of  scientific  classification,  wherever  such  an  ele- 
ment is  readily  perceivable  by  the  individuals  taught. 

To  assert  that  without  exception  all  psychological  order 
of  grouping  the  facts  included  in  the  school  curriculum 
must  be  logical  order  would  be  to  assume  not  only  that 
each  of  these  facts  has  in  common  with  other  facts  more 
or  less  that  is  effective,  but  that  every  fact  has  among  the 
various  elements  thus  common  and  effective  at  least  one 
which  through  psychological  instruction  can  be  readily 
perceived  by  the  great  majority  of  individuals  taught. 
While  this  assumption  is  quite  unnecessary  to  the  present 
argument,  experience  has  abundantly  demonstrated  that 
as  a  rule  such  an  element  can  be  found  in  the  various  facts 
which  go  to  make  up  the  common  school  curriculum. 

If  a  given  logical  order  based  upon  an  essential  common 
element  seems  too  difficult  for  the  child,  before  condemn- 
ing it  as  unpsychological,it  would  be  well  to  know  whether 
the  attempt  to  impart  the  facts  so  grouped  has  been  made 
in  accordance  with  a  psychological  method  of  instruction. 
(See  Chapter  III.)  And  even  should  this  prove  to  be  the 
case,  the  failure  of  the  child  to  comprehend  them  is  no 
indication  that  logical  order  in  general  is  unpsychological, 
but  rather  that  in  a  particular  logical  order,  the  common 
element  can  not  be  readily  perceived  by  the  child  mind, — 
the  right  course  in  such  an  event  being  to  seek  to  arrange 
the  same  facts  in  an  order  no  less  logical,  but  in  which 
the  essential  common  element  can  be  more  readily  per- 
ceived. 

The  discussion  thus  far  has  made  clear  the  conditions 
necessary  to  the  psychological  sub-grouping  of  the  facts 
included  in  any  general  category. 

In  each  psychological  group, — 

i.  The  common  element  must  be  an  effective  one;  one 
whose  perception  in  the  mastery  of  one  fact  in  the  group 
will  insure  the  ready  mastery  of  every  other  fact. 


28 

2.  It  must  be  the  most  effective  one  which  through 
psychological  instruction  can  be  readily  perceived  by  the 
great  majority  of  particular  individuals  to  be  taught, 
unless  some  other  effective  element  thus  readily  perceiv- 
able is  the  basis  of  the  accepted  scientific  classification. 

3.  It  must  be  common  to  more  facts  than  is  any  other 
readily  perceivable  and  equally  effective. 

All  the  facts  of  a  general  category  that  must  be 
mastered  by  the  child,  having  been  arranged  so  far  as  pos- 
sible in  psychological  groups,  it  is  evident  that  the  psycho- 
logical order  of  teaching  these  groups  will  depend  upon 
the  relative  effectiveness  of  their  common  elements,  the 
relative  readiness  with  which  their  common  elements  are 
perceived  by  particular  individuals  taught,  and  the  rela- 
tive number  of  facts  which  they  include.  When,  as  is 
usually  the  case,  mastery  of  the  effective  element  of  one 
group  is  a  necessary  condition  to  that  of  the  effective  ele- 
ment of  another,  or  when,  more  rarely,  the  effectiveness, 
perceivability  and  inclusiveness  of  the  elements  common  to 
the  various  groups  increase  or  diminish  together,  psycho- 
logical order  is  readily  determinable.  But  when,  as  may 
possibly  occur,  among  a  number  of  unrelated  groups,  the 
common  element  of  one  has  relatively  the  highest  effec- 
tiveness, of  another  the  readiest  perceivability,  and  of  a 
third  the  greatest  inclusiveness,  it  is  not  so  easy  to  deter- 
mine which  of  the  three  should  be  taught  first.  Each 
group,  it  may  be,  can  be  readily  mastered  by  the  child, 
but  should  they  vary  considerably  in  the  readiness  with 
which  they  can  be  mastered,  and  not  too  widely  in  the 
number  of  facts  which  they  include,  it  is  probably  more 
psychological  to  teach  first  the  one  which,  whether  through 
the  higher  effectiveness  of  its  common  element  or  through 
the  greater  readiness  with  which  that  element  is  perceiv- 
able, can  be  mastered  most  readily.  If  on  the  contrary 
one  should  include  many  more  facts  than  the  others,  being 
effective  and  readily  mastered,  though  not  most  readily 


29 

mastered,  it  should  be  taught  first,  tending  as  it  does  to 
the  readier  mastery  of  the  whole. 

Where,  as  will  usually  be  the  case  with  psychological 
groups,  the  groups  themselves  have  elements  in  common 
they  should  be  grouped  together  in  their  turn,  under  the 
same  conditions  as  the  single  facts  which  they  themselves 
include,  it  being  remembered  that  here,  the  readier  mastery 
of  some  one  group  must  be  subordinated  to  the  ready 
mastery  of  the  whole  collection  of  groups  of  which  it  is 
a  part. 

It  is  fortunate  that  this  general  discussion  of  psycho- 
logical order  has  not  involved  the  determination  of  what 
knowledge  is  of  most  worth.  Whether  or  not  physiology 
should  be  taught  before  number  or  aesthetics  before  read- 
ing, it  is  not  for  psychology  to  determine.  But  whatever 
the  sequence  of  the  general  branches  of  study — utilitarian, 
ethical,  or  historic,  there  will  be  few  among  the  facts 
selected  as  necessary  to  the  child,  that  can  not  be  included 
in  some  psychological  group.  Relative  usefulness,  char- 
acter building,  or  culture  epoch  may  demand  that  some 
individual  fact  shall  be  taught  before  the  psychological 
group  of  which  it  would  otherwise  form  a  part,  or  that 
some  logical  group  shall  be  taught  before  the  effective 
element  common  to  its  members  can  be  readily  mastered 
by  the  child,  but  after  all  such  demands  have  been  met, 
there  is  an  order  in  which  the  facts  thus  included  from 
year  to  year  in  the  school  curriculum  can  be  most  readily 
mastered  by  a  majority  of  the  pupils  taught,  and  this 
order  is  psychological. 

However  great  the  variation  in  our  courses  of  study, 
the  facts  which  are  taught  in  school  are  all  included  in 
some  general  classification.  They  are  grouped  together 
into  branches  of  study  in  each  school  curriculum,  and 
taught  for  the  most  part  with  the  aid  of  separate  text- 
books. Even  in  the  primary  grades  they  are  grouped  as 
number-work,  nature-study,  and  the  like.  The  common 
element  upon  which  this  general  classification  is  based 


30 

soon  comes  to  be  recognized — though  perhaps  somewhat 
vaguely — by  even  the  youngest  pupil.  He  is  not  able  to 
tell  what  it  is,  unless  the  teacher  has  pointed  it  out  or  de- 
noted it  by  some  suggestive  term,  but  he  will  not  confuse 
a  number-fact  with  his  reading  or  a  sentence  with  his 
number, — unless  "correlation"  has  quite  unified  his  mental 
content.  At  the  point,  however,  when  on  successive  sub- 
division, the  common  element,  however  effective,  is  not 
readily  perceivable  or  where  being  readily  perceivable  in- 
struction does  not  lead  the  learner  to  perceive  it,  logical 
order  becomes  of  no  avail,  except  in  so  far  as  it  prevents 
the  instructor  from  omitting  any  of  the  necessary  facts. 
To  the  learner  each  fact  stands  by  itself,  unrelated  to  other 
facts,  and  is  perceived  by  him  without  being  well  apper- 
ceived.  It  was  this  misuse  of  logical  order  that  char- 
acterized the  so-called  formal  instruction  which  here  and 
there  still  exists  in  American  schools.  To  the  influence 
of  German  pedagogy  and  more  latterly  to  the  populariza- 
tion of  psycho-genesis — especially  in  the  form  of  "child 
study,"  has  been  largely  due  the  reaction  against  formal 
instruction,  which  is  now  almost  everywhere  triumphant. 
Unfortunately,  however,  iconoclasm  here  as  elsewhere  the 
companion  of  reform,  has  shattered  the  old  idol  into  a 
thousand  fragments,  and  thereby  too  often  substituted 
for  logical  order  in  elementary  instruction,  the  study  of 
isolated  facts. 


II.  THE    PSYCHOLOGICAL    ORDER   OF   TEACHING   THE 
FUNDAMENTAL  SUMS  AND  DIFFERENCES. 

(I)  THE    REACTION    AGAINST    FORMALISM    IN    THE 
TEACHING  OF  ELEMENTARY  NUMBER. 

The  changes  which  have  taken  place  in  the  study  of 
arithmetic  during  the  past  fifty  years  well  illustrate  this 
swing  of  the  pedagogic  pendulum.  But  a  generation  or  so 
ago  formalism  reigned  supreme.  Beginners  were  taught 


notation  and  numeration  to  billions,  before  they  began  ad- 
dition ;  the  addition  of  columns  long  and  broad,  before  they 
studied  subtraction,  and  so  on  and  on  through  a  logical 
order  which  they  were  rarely  led  to  comprehend,  counting 
on  fingers,  learning  by  rote,  and  working  by  rule,  they 
plodded  from  multiplication  table  to  the  rule  of  three.  No 
one  will  claim  that  to  the  majority  of  those  taught  such 
teaching  was  psychological,  but  notwithstanding  its 
theoretic  faults,  the  reformer  is  occasionally  non-plussed 
by  the  undeniable  fact  that  it  turned  out  a  host  of  strong 
mathematicians. 

The  reason  is  not  far  to  seek.  All  that  was  lacking, 
where  the  element  common  to  each  group  was  an  effective 
common  element,  was  that  it  should  be  comprehended  by 
the  individuals  taught.  And  many  a  plodder  did  compre- 
hend even  as  he  labored,  while  many  another  came  with 
the  aid  of  a  maturer  mind,  to  ultimately  comprehend  what 
had  been  mechanically  but  permanently  drilled  into  his 
memory.  To  such  the  old  formal  instruction  was  either 
psychological  or  became  so.  Not  so,  however,  with  the 
great  majority,  stupefied  by  a  logic  they  could  not  under- 
stand, and  arrested  in  mental  development  by  what  was  to 
them  for  the  most  part  the  mechanical  memorizing  of  dis- 
connected facts.  With  the  more  or  less  individual  instruc- 
tion of  the  old  district  school,  the  formal  grouping  of  the 
facts  of  number  was  undoubtedly  helpful  to  the  few.  But 
as  no  educational  plan,  through  its  helpfulness  to  a  minor- 
ity can  find  justification  for  its  application  to  all,  educa- 
tional leaders — largely  influenced  by  the  German  reform- 
ers— began  to  protest  against  the  mechanical  mastery  of 
fact  and  rule,  and  to  urge  in  its  place  the  teaching  of 
"intellectual  arithmetic."  In  school  journal  and  teachers' 
meeting  they  persistently  insisted  that  the  pupil  should  not 
recite  rules  "by  heart  without  thinking  or  considering 
thereon,"  that  in  the  solution  of  problems  he  should 
examine  the  "question,"  and  not  his  stock  of  rules ;  that, 
in  fact,  there  should  be  no  rules  at  all,  but  in  their  stead 


32 

a  few  general  processes  which  would  furnish  the  key  to 
all  solutions.  Thus  arithmetical  analysis  and  mental 
arithmetic  came  to  find  a  place  in  the  curriculum, — substi- 
tuting for  rigidity  of  rule  a  no  less  rigidity  in  demonstra- 
tion, but  necessitating  the  teaching  of  the  subjectively 
simple,  before  that  of  the  subjectively  complex.  The 
triumph  of  reform,  even  yet  by  no  means  complete,  was 
not  won  without  a  struggle.  The  embattled  host  of  tradi- 
tion long  continued  to  hold  its  own  against  the  cham- 
pions of  reason.  The  educational  literature  of  the  period 
is  rich  in  material  for  a  history  yet  to  be  written,  of  this 
stage  in  the  old  warfare  between  "Trojan"  and  "Greek." 
But  while  here  and  there  may  be  found  a  locality  where 
every  rule  has  been  hurled  from  its  sacred  niche,  there  has 
hardly  yet  resulted,  as  feared  by  a  grandiloquent  conserva- 
tism, a  generation  of  pupils,  who  independent  alike  of 
teacher  and  text-book,  "never  soar  by  faith  to  regions  of 
inconceivable  glory,  but  ever  plod  on  by  the  dim  light  of 
reason."16 

The  only  innovation  that  need  be  here  discussed  is  the 
change  which  took  place  in  the  order  of  teaching  the 
elementary  facts  of  number.  Largely  through  the  influ- 
ence of  Colburn  and  Emerson,  the  fundamental  facts  of 
addition,  subtraction,  multiplication  and  division  began  to 
be  taught  before  the  study  of  numeration  and  notation,  or 
operation  with  numbers  greater  than  10  or  20.  Later 
came  the  introduction  of  the  Grube  method,17  involving 
another  order  no  less  logical  than  the  old  but  which  has 
for  its  end  the  exhaustion  of  the  "simpler"  of  the  funda- 
mental facts  before  the  study  of  those  objectively  more 
complex.  No  effort  was  made  to  so  group  the  facts  that 
the  mastery  of  one  might  involve  the  ready  mastery  of 
many.  On  the  contrary  ready  mastery  was  to  come 
through  the  method  by  which  each  individual  fact  was  to 
be  imparted.  Even  to-day  while  the  Grube  method  is 
rapidly  being  displaced  by  what  are  perhaps  its  betters, 
the  Grube  order  still  reigns  supreme  in  American 


33 

schools.18  Is  this  order  psychological?  If  not,  what  is 
the  psychological  order  of  teaching  the  elementary  sums 
and  differences  ? 

(II)  DESCRIPTION  OF  THE  VARIOUS  LOGICAL  ORDERS 
POSSIBLE  IN  TEACHING  THE  ELEMENTARY  SUMS 
AND  DIFFERENCES. 

The  terms  "sum"  and  "difference"  are  here  used  to 
denote  not  merely  the  results  of  certain  numerical  opera- 
tions, but  in  their  popular  signification,  as  denoting  the 
operations  themselves,  including  their  results.  For 
example,  3  +  2  =  5  is  called  a  sum,  and  5  —  3  =  2,  a  dif- 
ference. An  "elementary"  sum  is  one  formed  by  the 
addition  of  one  digit  to  another ;  a  "secondary"  sum,  one 
in  which  one  or  both  of  the  numbers  added  are  greater 
than  nine.  The  "inversion"  of  a  sum  only  differs  from 
the  sum  itself,  in  that  the  number  to  which  addition  is 
made  has  changed  place  with  the  number  to  be  added. 
An  "elementary"  difference  is  one  formed  by  the  sub- 
traction of  one  digit  from  another;  a  "secondary"  differ- 
ence, one  in  which  either  subtrahend  or  remainder  is 
greater  than  nine.  An  "alternation"  is  a  difference  in 
which  the  remainder  has  changed  places  with  the  subtra- 
hend. All  sums  and  differences,  whether  elementary  or 
secondary,  whose  mastery  is  necessary  to  ready  operation, 
may  be  regarded  as  "fundamental."  These  fundamental 
sums  and  differences  together  with  the  fundamental  prod- 
ucts, quotients,  etc.,  are  commonly  known  as  "the  number 
facts." 

As  the  secondary  facts  are  readily  derivable  from  those 
that  are  elementary,  and  the  identity  of  result  in  sum  and 
inversion,  or  the  interchange  of  result  in  difference  and 
alternation,  are — as  will  be  presently  demonstrated — 
readily  perceivable  by  the  great  majority  of  children,  it  is 
unnecessary  for  the  present  to  consider  the  arrangement 
of  all  fundamental  facts  that  are  not  elementary,  or  of 
elementary  facts  that  are  inversions  or  alternations. 


34 

Again,  while  it  has  not  been  taken  for  granted  that  the 
elementary  differences  should  not  be  taught  contempor- 
aneously with  the  elementary  sums,  it  will  better  serve  the 
purposes  of  the  discussion  to  fix  the  psychological  order 
for  the  teaching  of  the  latter,  before  considering  whether 
the  former  should  be  taught  thus  contemporaneously,  at  a 
later  period,  or  not  at  all. 

Can  the  elementary  sums  and  differences  be  grouped  in 
more  than  one  logical  order?  If  so,  what  are  these  logi- 
cal orders,  and  which  among  them  is  the  psychological 
order  for  the  majority  of  children  on  first  entering  the 
lowest  school  grade?  It  is  not  until  this  latter  question 
has  been  fully  determined  by  a  strict  application  of  the  test 
already  laid  down  that  we  can  judge  as  to  how  far  we  can 
utilize  the  knowledge  of  number  which  children  possess 
on  first  entering  school. 

In  building  up  a  group  of  different  elementary  sums  it 
is  evident  that  but  one  term  can  remain  constant.  If  the 
result — the  "sum"  in  the  stricter  sense  of  the  term — is  not 
varied,  both  the  number  to  be  added  and  that  to  which 
addition  is  to  be  made,  become  variables ;  if  either  of  the 
latter  is  kept  constant,  the  number  added  to  it  or  to  which 
it  is  added,  and  their  result,  must  be  variables.  Moreover, 
it  is  evident  that  if  a  system  of  grouping  is  to  include  all  of 
the  elementary  sums,  it  must  involve  the  addition  of  each 
digit  to  every  other  digit, — though  not  necessarily  in  the 
usual  digit  order.  Hence  only  three  distinct  sets  of  groups 
of  elementary  sums  are  possible :  ( i )  Those  in  which  the 
"result"  is  constant;  (2)  those  in  which  the  digit  to  which 
addition  is  to  be  made,  is  constant,  and  (3)  those  in  which 
the  number  to  be  added  is  constant.  In  each  set  every 
digit  must  be  the  constant  in  a  group  containing  all  the 
elementary  sums,  which  can  be  formed  by  the  variation 
of  the  other  two  terms.  If  the  order  of  teaching  the 
elementary  sums  is  to  be  psychological  in  the  highest 
degree  it  must  be  determined : 

i.  Whether  one  system  of  grouping  is  psychological 


35 

throughout  as  compared  with  either  of  the  others,  or 
whether  psychological  order  necessitates  the  use  of  groups 
from  more  than  one  system; 

2.  The  psychological  order  of  the  groups  selected. 

3.  The  psychological  order  of  the  facts  within  each 
group. 

For  convenience  in  the  discussion,  the  forty-five 
elementary  sums  not  inversions  are  given  below,  grouped 
according  to  each  system  and  following  the  order  of  the 
digits. 

TABLE  OF  THE   ELEMENTARY  SUMS  ARRANGED  UNDER 
EACH  OF  THE  THREE  LOGICAL  SYSTEMS  OF  GROUPING. 

A.  The  Grouping  Together  of  the  Sums  Obtained  by  the  Addition  of 

the  Several  Digits  in  Order  to  Each  Successive  Digit. 

1.  1  &  1,  1  &  2,  1  &  3,  1  &  4,  1  &  5,  1  &  6,  1  &  7,  1  &  8,  1  &  9; 

2.  2  &  2,  2  &  3,  2  &  4,  2  &  5,  2  &  6,  2  &  7,  2  &  8,  2  &  9; 

3.  3  &  3,  3  &  4,  3  &  5,  3  &  6,  3  &  7,  3  &  8,  3  &  9; 

4.  4  &  4,  4  &  5,  4  &  6,  4  &  7,  4  &  8,  4  &  9; 

5.  5  &  5,  5  &  6,  5  &  7,  5  &  8,  5  &  9; 

6.  6&6,  6&7,  6&8,  6&9; 

7.  7&7,  7&8,  7&9; 

8.  8&8,  8&9; 

9.  9  &  9. 

B.  The  Grouping  Under  Each  Successive  Digit  of  the  Sums  Whose 

Results  Are  Equal  to  It. 


1.  1& 

2.  2& 

3.  3& 

4.  4& 

5.  5& 

6.  6& 

7.  7& 

8.  8& 


;  10.  9  &  2,8  &  3,7  &  4,6  &  5; 

;  11.  9  &  3,8  &  4,7  &  5,6  &  6; 

,  2  &  2;  12.  9  &  4,  8  &  y,  7  &  6; 

,3&2;  13.  9  &  5,8  &  6,7  &  7; 

,4&2,  3&3;  14.  9  &  6,8  &  7; 

,  5&2,  4&3;  15.  9  &  7,8  &  8; 

,6&2,5&3,4&4;  16.  9&8; 

,  7&2,  6&3,  5&4;  17.  9  &  9. 


9.  9  &  1,  8  &  2,  7  &  3,  6  &  4,  5  &  5; 

C.  The  Grouping  Together  of  the  Sums  Obtained  by  the  Successive 
Addition  of  Each  Digit,  First  to  Itself,  Then  to  Each  Inferior 
Digit. 

1.  1  &  1,  2  &  1,  3  &  1,  4  &  1,  5  &  1,  6  &  1,  7  &  1,  8  &  1,  9  &  1; 

2.  (1)2&2,4&2,6&2,8&2;   5.  (1)  5&5;   6.  (6) 8.  (3) 9.  (4)  — 

(2)3&2,5&2,7&2,9&2;       (2)  6&5;   7.  (1)  7&7;        (4) (5)  — 


36 

3.(1)3&3,6&3,9&3;  (3)  7&5;  (2)8&7;       (5) (6) — 

(2)4&3,7&3;  (4)  8&5;  (3)9&7;        (6) (7) — 

(3)5&3,8&3;  (5)  9&5;  (4)-             (7) (8) — 

4 .  (1)  4&4, 8&4 ;  6.  (1)  6&6 ;       (5) (8) (9)  — 

(2)5&4,9&4;  (2)  7&6;       (6) 9.  (1)  9&9; 

(3)6&4;  (3)8&6;       (7) (2) 

(4)  7&4;  (4)  9&6;  8.  (1)  8&8;       (3) 

(5) (2)  9&8; 

(III)  COMPARISON  OF  THE  THREE  SYSTEMS  WITH  A 
VIEW  TO  THE  DETERMINATION  OF  THE  PSYCHO- 
LOGICAL ORDER. 

Is  one  system  of  grouping  more  psychological  through- 
out than  is  either  of  the  others  ? 

To  determine  this,  the  usual  test  must  be  applied  to  the 
various  groups  resulting  from  the  completed  development 
of  each  system.  Have  the  groups  of  any  one  system,  as 
compared  with  those  of  the  other  two,  common  elements : 

1.  More  effective? 

2.  More  readily  perceivable  by  those  to  be  taught  ? 

3.  Common  to  a  greater  number  of  the  elementary 
sums? 

It  must  be  steadily  held  in  mind  that  among  these  tests 
that  of  effective  identity  is  of  the  highest  comparative 
importance, — that,  if  the  mastery  of  one  individual  of  the 
group  is  not  the  key  to  the  mastery  of  all,  the  grouping 
can  be  psychological  only  in  so  far  as  it  insures  the  teach- 
ing of  facts  subjectively  simple  before  those  relatively 
complex. 

Certain  numbers  are  to  be  firmly  associated  together  as 
individual  number  facts.  If  any  considerable  number  of 
these  facts  can  be  so  grouped  together  that  the  mastery  of 
one  involves  the  mastery  of  all,  their  partial  identity  may 
be  regarded  as  an  effective  identity.  The  group  is  psycho- 
logical, however,  only  when  this  effective  identity  is 
readily  perceivable  by  a  majority  of  the  individuals  who 
are  to  thus  firmly  associate  the  digits  together  as  facts. 
The  more  effective  this  identity,  the  more  readily  it  is  per- 


37 

ceivable,  and  the  greater  the  number  of  facts  which  possess 
it,  the  more  psychological  the  resulting  group. 

i.  Why  the  Grouping  Together  of  the  Sums  Obtained 
by  the  Addition  of  the  Several  Digits  in  Order  to  Each 
Successive  Digit  is  Not  the  Psychological  System  of 
Grouping. 

This  grouping  which  results  from  the  addition  of  all  the 
digits  to  each  digit  as  a  constant,  prevailed  in  American 
schools  from  the  time  of  Colburn  until  the  Grube  method 
became  popular.  It  is  found  for  example  in  such  well- 
known  text-books  as  those  of  Emerson  (1838),  Green- 
leaf  (1851),  and  Davies  (1862),  as  well  as  in  many 
others.  The  constant  addend  to  which  addition  is  made 
and  the  fact  that  when  the  sums  are  taken  in  the  order  of 
the  digits,  each  result  is  greater  by  one  than  that  immedi- 
ately preceding  it,  are  the  elements  common  to  all  the 
facts  of  a  given  group,  not  considered  in  relation  to  other 
groups.  There  is  nothing,  however,  in  the  fact  that  the 
pupil  has  added  one  number  to  6,  to  aid  him  in  giving 
the  result  obtained  by  the  addition  of  another  number  to 
6,  unless  he  successively  gives  all  the  facts  which,  when 
taken  in  order,  intervene  between  them.  Thus,  knowing 
that  6  and  2  are  8,  he  can  know  that  6  and  5  are  n,  by 
repeating  to  himself,  6  and  2  are  8,  6  and  3  are  9,  6  and  4 
are  10,  6  and  5  are  n,— an  operation  which  was  usually 
performed  with  the  aid  of  finger-counting. 

But  this  knowledge  is  not  due  to  the  grouping.  The 
result  may  be  just  as  readily  obtained  for  the  isolated 
fact.  The  method  employed  is  merely  the  counting  out 
of  the  sum,  either  mentally  or  with  the  aid  of  objects.  It 
is  evident  then  that  the  elements  common  to  the  various 
groups  of  this  system,  considered  within  themselves,  are 
not  effective.  When,  however,  the  group  formed  by  the 
addition  of  the  digits  to  5  has  been  thoroughly  mastered 
before  the  study  of  the  6  group  begins,  the  facts  included 
in  the  latter  have  an  effective  common  element, — in  that 


any  fact  in  the  6  group  will  have  a  result  greater  by  one 
than  that  of  the  corresponding  fact  in  the  5  group.  If  5 
and  5  are  10,  6  and  5,  or  i  and  5  and  5  will  be  1 1.  This, 
however,  is  not  so  readily  perceivable  by  the  child  as  is  the 
fact  that  if  5  and  5  are  10,  5  and  5  and  i,  or  5  and  6  will  be 
1 1 — which  is,  as  will  be  seen,  all  that  it  is  necessary  for  him 
to  perceive  in  the  third  system  of  grouping.  In  the  former 
case  he  must  be  led  to  see  that  the  addition  of  i  to  the  5, 
is  the  addition  of  I  to  the  5  and  5 ;  in  the  latter,  that  the 
addition  of  the  i  to  the  5  and  5  involves  the  addition  of 
the  i  to  the  5.  As  the  result  can  be  obtained  inductively 
only  by  first  adding  the  i  to  the  5  and  5,  or  the  5  and  5  to 
the  i,  it  is  evident  that  the  latter  method  of  derivation  will 
be  more  readily  mastered  by  the  child.  It  is  the  algebraic 
5  and  5  and  i  compared  with  i  and  (5  and  5).  The  5 
must  be  added  to  the  5  before  the  child  perceives  that  the 
addition  of  the  i  results  in  the  n.  It  is  after  this  that  he 
must  perceive  that  the  addition  of  I  and  5,  or  of  5  and  I, 
is  the  addition  of  6.  This,  too,  is  more  readily  seen  by  the 
latter  plan,  where  it  has  not  been  necessary  to  overempha- 
size the  addition  of  the  5  to  the  5.  Under  this  system  of 
grouping,  which  it  may  be  convenient  to  designate  as  the 
Colburn,  there  are  nine  groups  containing  from  nine  to 
one  elementary  facts. 

2.  Why  the  Grouping  Under  Each  Successive  Digit  of 
the  Sums  Which  Have  It  for  Their  Result  is  Not  the 
Psychological  System  of  Grouping. 

This  is  of  course  the  Grube  system  of  grouping,  limited 
to  a  certain  class  of  fundamental  facts.  For  thirty  years 
or  more  it  has  been  gradually  superseding  the  Colburn 
plan,  until  to-day  there  are  few  if  any  elementary  arithme- 
tics that  do  not  follow  it.  The  element  common  to  the 
sums  composing  one  of  its  groups  is  their  result,  and  that 
alone,  unless,  as  is  not  done  in  the  Grube  method  of  teach- 
ing, the  result  is  considered  in  its  relation  to  that  of  the 


39 

group  immediately  below.  In  this  latter  case,  it  may  be 
regarded  as  one  greater  for  each  sum  than  the  result  for 
the  corresponding  sum  in  the  lower  group, — a  common 
element  readily  enough  perceived  after  it  has  been  ob- 
tained, but  hardly  effective,  in  that,  for  example,  4  and 
2  are  known  to  be  6  because  5  and  i  are  6.  The  child 
could  only  recognize  an  unknown  sum  as  belonging  to  the 
6  group  through  the  fact  that  another  sum,  one  of  whose 
addends  was  identical  with  one  addend  of  the  unknown, 
and  the  other  less  by  one  than  the  other  addend  of  the  un- 
known, had  5  for  its  result, — a  generalization  which  once 
mastered  would  be  applicable  to  all  groups,  but  whose 
mastery  is  at  least  sufficiently  difficult  to  have  prevented 
any  attempt  to  utilize  it  by  the  disciples  of  Grube.  Under 
the  Grube  order  each  elementary  sum  has  been  taught  as 
a  fact  separate  and  distinct  from  other  elementary  sums, 
— except  in  so  far  as  with  each  sum  has  been  taught  its 
inversion,  which  practice  with  that  of  simultaneously 
teaching  the  corresponding  elementary  differences,  prod- 
ucts and  quotients,  need  not  now  be  considered.  As  a 
grouping  is  possible  in  which  the  mastery  of  one  elemen- 
tary sum  in  a  group  will  insure  the  ready  mastery  of  all, 
the  Grube  order  is  obviously  unpsychological. 

It  has  been  continually  urged  in  its  favor  that  it  in- 
sures the  teaching  of  the  "simpler"  facts  before  that  of  the 
"more  complex."  By  it,  it  is  claimed  the  children  are  not 
confused  by  the  introduction  of  the  more  difficult  numbers, 
before  the  easier  ones  are  mastered.  Unfortunately  for 
this  argument,  however,  the  great  majority  of  children  on 
entering  school,  are  already  familiar  with  the  names  of  all 
the  digits.  In  fact  many  a  teacher  of  the  Grube  system 
has  found  herself  forced  into  the  absurd  position  of  wish- 
ing that  her  children  knew  less.  But  granting  that  as 
far  as  number  is  concerned,  the  mind  of  the  beginner  is  a 
tabla  rasa,  it  by  no  means  follows  that  by  teaching  the 
sums  as  isolated  facts,  the  Grube  order  presents  first 
those  that  are  the  "easiest"  for  the  child.  No  fact  has 


40 

been  better  demonstrated  by  psychologists  than  that  the 
objectively  simple  may  be  subjectively  complex,  and  the 
objectively  complex,  the  subjectively  simple.  It  has  not, 
for  example,  been  satisfactorily  shown,  that  the  associa- 
tion 5  and  4  is  more  difficult  for  the  child  than  is  that  of 
2  and  3.  Even  aside  from  this,  objectively  considered,  it 
would  be  hard  to  express  in  mathematical  terms  the  differ^ 
ence  in  "simplicity"  between  4  and  2  and  8  and  i.  And 
yet,  it  is  only  in  so  far  as  through  it  the  beginner  may  be 
first  taught  those  individual  facts,  which  he  can  most 
readily  master,  that  the  Grube  order  is  in  any  respect  an 
improvement  upon  the  most  arbitrary  order  of  teaching 
the  elementary  sums  which  will  include  them  all,  especially 
as  it  divides  them  into  seventeen  groups,  where  the  Col- 
burn  groups  them  in  nine. 

3.  Why  the  Grouping  Together  of  the  Sums  Obtained 
by  the  Successive  Addition  of  Each  Digit,  First  to  Itself 
Wid  Then  to  Each  Inferior  Digit,  is  Psychological. 

There  remains  to  be  considered  the  system  in  whose  suc- 
cessive groups  the  constant  term  is  the  digit  added.  While 
it  has  never  been  used  so  generally  as  the  two  already  dis- 
cussed, it  has  occasionally  found  its  champions,  as,  for 
example,  when  in  the  early  sixties,  Felter  and  Eaton  used 
it  in  their  elementary  arithmetics.  In  it,  the  element  com- 
mon to  the  facts  included  in  a  given  group  is  the  number 
to  be  added  combined  with  the  fact  that  the  result  is 
always  one  greater  than  the  corresponding  result  in  the 
group  immediately  below.  Unlike  that  just  considered, 
this  common  element  is  highly  effective.  For  example, 
the  pupil  who  is  once  led  to  see  that  5  and  3  are  8  because 
knowing  that  5  +  2  =  7,  he  will  know  that  5  +  2+1=8, 
will  readily  come  to  derive  for  himself  the  result  of  every 
other  elementary  sum  involving  the  addition  of  3.19  That 
the  element  thus  effective  was  readily  perceivable  by  the 
great  majority  of  the  pupils  in  the  first  school  grade  of 
one  representative  locality,  will  be  presently  demonstrated. 


Meanwhile,  being  effective, — as  the  common  element  in 
the  Grube  group  is  not, — being,  if  not  more  effective,  at 
least  far  more  readily  perceivable,  than  that  of  the  Colburn 
series,  it  is  plain  that  the  system  of  which  it  is  the  basis, 
with  its  nine  groups,  is  the  psychological  system. 

The  pupil  who  has  once  made  the  simple  generaliza- 
tion that  one  added  to  a  number  always  results  in  the 
number  next  above  it  in  the  scale,  will  readily  derive  for 
himself  with  the  aid  of  proper  instruction — which  in 
its  primal  sense  means,  after  all,  the  right  arrangement 
of  subject  matter — group  by  group,  all  of  the  elementary 
sums.  The  thorough  mastery  of  one  group  insures  the 
ready  derivation  of  the  next.  Knowing  all  the  elementary 
sums  obtainable  by  the  addition  of  four,  he  will  readily 
derive  and  for  the  most  part,  quite  abstractly, — all  of 
those  obtained  by  the  addition  of  4  and  I  or  5. 

As  this  is  the  case  with  all  the  groups  and  the  deriva- 
tion of  each  group  depends  upon  the  thorough  mastery  of 
that  immediately  preceding,  it  follows  that  the  system 
under  discussion,  being  psychological  throughout  as  com- 
pared with  either  of  the  others,  it  is  unnecessary  to  use 
groups  from  more  than  one  system. 

(IV)  THE   DETERMINATION   OF  THE   PSYCHOLOGICAL 
ORDER  OF  THE  GROUPS  SELECTED  AS  A  RESULT  OF 
THIS  COMPARISON. 

One  group  being  dependent  for  its  derivation  upon  the 
thorough  mastery  of  the  group  immediately  below,  the 
psychological  order  of  the  groups  within  the  system  is 
naturally  determined :  First,  the  elementary  sums  formed 
by  the  addition  of  i ;  second,  those  formed  by  the  addition 
of  2,  etc.,  to  those  formed  by  the  addition  of  9. 

(V)  THE    DETERMINATION    OF    THE    PSYCHOLOGICAL 
ORDER  FOR  THE  FACTS  WITHIN  EACH  GROUP. 

The  elementary  sums  are  not  only  to  be  independently 
derived  by  the  pupils,  but  so  thoroughly  memorized  that 
the  result  will  be  instantly  called  to  mind,  when  the 


42 

addends  are  seen  or  heard.  This  ready  recall  can  be  con- 
sequent only  upon  repetition, — a  repetition  either  abstract 
and  mechanical,  or  varied  and  concrete, — of  the  isolated 
fact  or  of  the  facts  in  relation  to  each  other.  For 
example,  4  and  2  are  6,  once  derived,  can  be  memorized 
either  by  repeating  concrete  applications — such  as  4  cats 
and  2  cats,  or  4  cents  and  2  cents,  or  by  a  mechanical  repe- 
tition of  the  abstract  4  +  2  =  6.  Neither  plan  is  wholly 
satisfactory.  The  first  is  uneconomical,  in  that  too  much 
time  is  taken  in  getting  the  necessary  amount  of  repeti- 
tion. Though  properly  used,  it  appeals  to  the  child's 
interest,  and  though  applying  the  facts,  it  is  a  necessary 
form  of  the  number  drill,  it  is  not  the  form  of  repetition 
which  will  result  in  the  memorizing  of  the  facts  with  a 
minimum  expenditure  of  time  and  effort.  The  second  is 
economical  until  the  attention  of  the  pupil  is  lost  through 
his  lack  of  interest  and  the  fatigue  arising  from  the  un- 
varied repetition.  It  can  be  occasionally  used,  to  fix  in 
the  pupil's  memory  some  individual  fact,  but  is  manifestly 
unfitted  for  the  regular  drill  upon  the  facts  of  a  given 
group. 

Again  the  4  and  2  are  6  can  be  memorized  in  an  abstract 
series  where  it  is  associated  with  other  facts, — as  in  the 
"addition  table."  In  the  case  of  the  tables  as  usually 
arranged,  the  pupil  mechanically  commits  to  memory  a 
series  of  facts, — which  he  may  or  may  not  have  derived 
for  himself, — in  such  a  way  that  when  a  fact  is  demanded 
out  of  its  usual  order,  it  often  can  be  recalled  only  through 
a  running  over  of  that  part  of  the  table  immediately  pre- 
ceding it,  to  the  end  that  a  false  and  unnecessary  associa- 
tion may  be  utilized.  Nevertheless  the  table  has  always  been 
popular  with  the  pupils,  and  much  of  the  disrepute  into 
which  it  has  fallen  among  the  "more  advanced"  teachers, 
is  due  to  their  naturally  but  unnecessarily  associating  it 
with  the  irrational  instruction  of  which  it  so  often  formed 
a  part,  and  their  erroneously  assuming  that  the  child  finds 
the  monotonous  repetition  as  unpleasant  as  does  the  adult. 


43 

It  is  this  very  rhythmical,  sing-song  movement  that  gives 
pleasure  to  the  child;  it  is  this  combined  with  the  joy  of 
achievement  that  makes  him  love  to  count.21  If  the 
fundamental  sums  could  be  so  arranged  that  the  associa- 
tion formed  by  repeating  them  in  succession  would  be  an 
essential  association  of  result  with  its  addends,  rather 
than  the  false  one  of  sum  with  sum,  the  repetition  of  these 
facts  in  tables  would  be  the  psychological  way  of  memo- 
rizing them,  because  it  would  be  the  readiest  way,  and  the 
way  most  pleasant  to  the  child.  Now  it  is  just  such  an 
arrangement  that  results  from  the  sub-grouping  of  tha 
sums  formed  by  the  successive  addition  of  a  given  digit, 
into  those  in  which  it  is  successively  added  beginning 
with  itself,  and  those  in  which  it  is  successively  added  be- 
ginning with  each  inferior  digit, — that  is,  the  arrangement 
which  is  based  upon  counting  by  2's  from  2  and  from  I ; 
by  3's,  from  3,  from  i  and  from  2,  etc.20  Of  course  this 
drill  includes  not  only  the  elementary  sums  not  inversions, 
but  all  the  fundamental  addition  facts.  Two  and  4,  though 
identical  in  its  result  with  4  and  2,  and  14  and  2,  though 
but  4  and  2  and  10,  must  be  as  readily  recalled  as  the  4 
and  2  itself. 

Knowing  as  the  children  do,  that  they  are  counting  say 
by  4's  from  2,  as  they  repeat  "6,  10,  14,  18,"  while  the 
teacher  interjects  "and  4?"  they  are  forming  the  associa- 
tion 6  and  4  are  10,  10  and  4  are  14,  etc.,  in  such  a  way 
that  the  series  once  mastered,  all  of  its  individual  facts 
are  mastered.  The  drill  is  therefore  economical.  While 
abstract  it  is  varied, — that  is,  it  is  not  the  mere  repeti- 
tion of  one  fact  again  and  again.  It  is  interesting  to  the 
children  from  their  love  of  counting,  due  not  alone  to  its 
pleasurable  rhythm,  but  as  they  master  successive  series, 
to  the  joy  of  achievement  as  well — the  satisfaction  arising 
from  the  knowledge  that  they  can  count  by  3's  and  4's 
instead  of  merely  by  I's  or  by  2's.21  Where  a  digit  is 
successively  added  to  itself,  the  way  is  opened  for  the 
ready  derivation  of  the  fundamental  products  and  quo- 


44 

tients.22  Finally,  the  only  numerical  knowledge  which 
the  great  majority  of  children  have  in  common  on  enter- 
ing school, — the  ability  to  more  or  less  mechanically  count 
by  i's,  within  narrow  limits, — is  utilised  from  the  very 
start.  It  is  with  counting  by  I's  that  the  work  begins. 

It  is  of  course  admitted  that  the  counting  is  "mechani- 
cal memory  work"  but  what  of  that,  if,  though  abstract, 
it  is  interesting  to  the  child,  and,  through  the  very  fact 
that  it  is  mechanical  the  readiest  way  of  fixing  firmly  in 
his  mind  facts  that  he  has  already  intelligently  derived. 

This  sub-grouping,  then,  of  the  psychological  groups 
is  psychological,  since  the  facts  are  so  arranged  that  as 
each  group  is  derived,  it  may  be  memorized  by  the  child 
with  a  maximum  of  readiness  and  interest. 

So  far,  it  is  only  those  of  the  fundamental  sums  which 
are  less  than  24,  that  have  been  psychologically  grouped, 
There  remains  a  number  of  groups  no  less  fundamental, 
whose  psychological  order  must  be  determined.  The 
sums  obtained  by  the  successive  addition  of  lo's  from  10; 
the  sums  obtained  by  the  addition  of  the  various  digits  to 
the  multiples  of  10  thus  derived;  the  sums  obtained  by  the 
additions  of  i  to  every  number  from  24  to  99, — necessary 
to  intelligent  counting  to  100, — while  so  related  to  each 
other  that  they  must  be  taught  in  the  order  in  which  they 
were  just  named,  should  have  their  place  in  the  general 
psychological  order  as  certainly  established,  as  that  of 
the  sums  already  discussed.  To  do  this  it  is  necessary  to 
assume,  what  later  will  be  convincingly  proven, — that  it  is 
as  easy  for  the  majority  of  children  to  add  ID'S  as  to  add 
apples,  and  far  easier  to  add  a  digit  to  30  or  40,  or  i  to  25 
or  48,  than  to  add  2  to  6  or  9.  This  being  the  case,  the 
psychological  order  for  these  groups  of  fundamental 
sums  is  immediately  after  the  i  sums  from  3  and  i  to  9 
and  i,  and — on  account  of  the  difficult  terms — n  and  12 
— before  the  i  sums  from  10  and  i  to  19  and  i.  Thus 
the  decimal  concept  is  formed  from  the  start,  and  a  large 
number  of  fundamental  facts  mastered  through  the  ready 


45 

mastery  of  a  common  element  necessary  to  the  compre- 
hension of  the  decimal  system. 

(VI)  THE  GENERAL  PSYCHOLOGICAL  ORDER  OF  TEACH- 
ING THE  FUNDAMENTAL  SUMS,  AS  THUS  DETER- 
MINED ON  A  PRIORI  GROUNDS. 

1.  Intelligent  counting  by  I's  to  10;  the  "one  sums" 
and  their  inversions. 

2.  Intelligent  counting  by  lo's  to  90;  the  "ten  mul- 
tiples" and  the  "ten  sums."     Place  value  to  lo's. 

3.  Intelligent  counting  both  by  the  addition  of  the 
successive  digits,  to  the  10  multiples  and  by  i's — (i) 
from  20  to  100;  (2)  from  10  to  20. 

4.  Intelligent  counting  first  to  12  and  then  to  24,  by 
2's,  3's,  4's,  5's,  etc.,  successively— (i)  from  2,  3,  4,  5, 
etc.,  respectively;  e.  g.  "4,  8,  12,"  or  "2,  4,  6,  8,  10,  12." 
(2)  Successively  from  all  other  digits  less  than  the  one  by 
which  the  counting  is  being  done;  e.  g.  "i,  5,  9,   13; 
2,  6,  10;  3,  7,  ii."     All  the  elementary  sums  formed  by 
such  successive  addition  of  each  digit,  being  with  their 
inversions,  thoroughly  mastered,  before  the  beginning  of 
the  drill  on  counting  by  the  digit  next  succeeding. 

(VII)  THE  DETERMINATION  OF  THE  PSYCHOLOGICAL 
ORDER  OF  TEACHING  THE  FUNDAMENTAL  DIFFER- 
ENCES. 

In  determining  the  psychological  order  of  teaching  the 
fundamental  differences,  it  is  necessary  to  consider  not 
only  the  system  of  grouping  in  which  they  can  be  most 
readily  mastered  by  the  individuals  taught,  as  a  series 
distinct  from  all  other  facts,  but  also  to  fix  the  relation 
which  their  teaching  should  hold,  to  that  of  the  funda- 
mental sums — it  being  assumed  for  the  present  that  the 
teaching  of  the  fundamental  products  and  quotients 
should  follow  after,  except  those  whose  derivation  is  inci- 
dental to  elementary  addition. 


46 

Should  the  fundamental  differences  be  taught  at  all? 
If  so,  should  they  be  taught  contemporaneously  with  the 
fundamental  sums,  or  after  the  fundamental  sums  have 
been  thoroughly  mastered?  Whether  they  should  be 
taught  as  distinct  from  the  elementary  sums,  or  in  other 
words,  whether  subtraction  should  be  taught  other  than 
as  a  particular  form  of  addition,  is  a  question  which  has 
not  as  yet  been  experimentally  settled.  A  priori,  there 
is  little  doubt  that  a  difference  can  be  just  as  accurately 
and  as  quickly  obtained  by  considering  its  units,  tens,  etc., 
as  the  numbers  to  be  added  to  the  corresponding  digits 
of  the  subtrahend  in  order  that  they  shall  equal  those  of 
the  minuend,  as  by  taking  them  to  be  the  direct  result  of 
the  successive  subtraction  of  the  corresponding  digits  of 
the  subtrahend  from  those  of  the  minuend.  But  while  as 
a  mechanical  operation,  the  former  may  be  as  satisfactory 
as  the  latter,  it  is  more  difficult  of  application.  For 
example,  an  individual  who  wishes  to  determine  the  re^ 
mainder  which  will  result  from  taking  9  from  17,  must 
know  that  it  is  the  number  which  added  to  9  will  give  17. 
This  he  can  learn  either  unintelligently  or  intelligently ;  he 
can  either  blindly  accept  the  fact  that  the  difference  is  the 
number  which  added  to  the  subtrahend  will  give  the  minu-< 
end,  or  accept  it  because  he  has  made  the  generalization 
that  the  minuend  being  the  sum  of  the  subtrahend  and 
some  other  number,  if  the  subtrahend  is  taken  away  the 
other  number  must  remain.  As  intelligent  operation  is 
here  assumed,  this  generalization  becomes  a  necessary 
condition  to  subtraction  "by  addition."  But  an  individual 
who  has  made  it  can  derive  at  will  all  of  the  fundamental 
differences.  The  customary  drill  in  the  rapid  subtrac- 
tion of  small  numbers,  would  therefore  be  coincident  with 
drill  upon  the  fundamental  differences.  In  view  of  this 
fact,  it  is  probably  better  to  consider  the  making  of  the 
generalization,  where  it  can  be  made,  as  the  psychological 
method  of  deriving  them,  rather  than  as  a  necessary  step 
in  a  mode  of  subtraction,  which  with  them  once  mastered, 
presents  no  advantages  over  that  in  general  use. 


47 

If  the  fundamental  differences  are  to  be  taught, 
another  question  arises — shall  they  be  taught  simultane- 
ously with  the  fundamental  sums,  or  after  the  funda- 
mental sums  in  whole  or  in  part  have  been  thoroughly 
mastered.  Against  the  former  alternative  is  the  fact, 
that — at  least  where  number  is  taught  to  children  on  their 
first  entering  school, — many  are  so  immature  in  their 
mental  development,  as  to  be  sadly  confused  by  the  intro- 
duction of  the  "less  facts"  while  they  are  striving  to 
master  the  sums.  In  favor  of  it  is  the  practice  existing 
in  many  schools  of  teaching  sums  and  differences,  inver- 
sions and  alternations,  together — the  only  psychological 
grouping  possible  under  the  Grube  system.  For  example, 
with  5  and  i  and  i  and  5  are  taught  6  —  i  and  6  —  5,  all 
four  of  the  facts  having  the  5  and  the  i  variously  associ- 
ated with  the  6.  But  this  is,  after  all,  merely  the  objective 
derivation  of  each  fundamental  difference,  aided  by  the 
generalization  already  discussed.  It  is  manifestly  far 
more  psychological  to  thus  derive  no  more  of  the  differ- 
ences than  are  necessary  to  the  making  of  the  generaliza- 
tion, by  which,  once  made,  all  can  be  derived  without  the 
aid  of  objects.  The  postponement  of  this  derivation  until 
after  the  first  groups  of  fundamental  sums  are  thoroughly 
mastered,  will  give  abundant  opportunity  for  all  who  are 
capable  of  making  the  generalization  to  be  led  to  make  it, 
and  at  the  same  time  will  allow  those  not  capable  of 
making  it,  and  therefore  in  the  stage  of  mental 
development  where  confusion  of  sum  and  difference  will 
be  most  puzzling, — to  devote  their  undivided  attention 
to  the  mastery  of  the  fundamental  sums.  If  after  those 
failing  to  make  the  generalization  had  thus  mastered  the 
sums,  they  would  be  compelled  to  objectively  derive  the 
individual  differences  with  no  aid  from  the  grouping, — 
especially,  if  the  number  thus  failing  was, — as  it  proved 
to  be,  a  considerable  fraction  of  the  whole, — the  teaching 
of  the  fundamental  differences  after  the  sums,  might  not 
be  psychological.  But  aside  from  the  possibility  of  hav- 


48 

ing  such  pupils  wait  for  the  study  of  subtraction  until  they 
are  more  mature  and  are  able  with  their  "brighter"  com- 
panions, to  derive  the  fundamental  differences  at  will,  or 
to  dispense  with  them  altogether, — the  differences,  like 
the  sums,  can  be  taught  in  psychological  order,  and,  by  the 
same  simple  mental  process,  one  group  after  another 
readily  derived. 

Where  pupils,  once  having  mastered  the  fundamental 
sums,  can  be  readily  led  to  make  and  to  apply  the  gen- 
eralization, that  whenever  one  of  two  numbers  is  sub- 
tracted from  their  sum,  the  other  remains,  so  far  as  the 
mere  derivation  of  the  fundamental  differences  is  con- 
cerned, the  order  may  be  purely  arbitrary.  But  it  is 
necessary,  not  only  that  a  difference  shall  be  derived  but 
that  its  minuend  and  subtrahend  shall  be  so  firmly  associ- 
ated with  it — that  is,  with  the  "difference,"  in  the  nar- 
rower sense, — that  perception  of  the  former  shall  instan- 
taneously call  up  the  latter.  Here,  as  in  the  case  of  the 
fundamental  sums,  counting  seems  to  furnish  the  most 
psychological  form  of  the  repetition  without  which  this 
firm  association  would  be  impossible. 

Where  pupils  who  have  mastered  the  fundamental 
sums  are  unable  thus  to  derive  the  fundamental  differ- 
ences, the  psychological  order  of  teaching  the  latter  can 
be  shown  by  precisely  the  same  line  of  reasoning, — to  be 
essentially  the  same — though  in  inverse  order, — as  that  in 
which  the  former  should  be  taught.  The  pupils  will  first 
learn  to  count  backwards  by  I's — say  from  24.  This 
will,  with  proper  instruction,  give  them  the  fundamental 
differences,  resulting  from  the  subtraction  of  i.  Those 
resulting  from  the  subtraction  of  2  will  then  be  obtained, 
by  the  successive  subtraction  of  I, — thus  18  —  i  —  i  or 
1 8  —  2 ;  those  resulting  from  the  subtraction  of  3,  from 
the  subtraction  of  i  from  the  corresponding  differences  of 
the  "2  group,"  etc. 

The  sub-grouping,  as  before,  will  be  such  as  to  make 
possible  counting  backwards  by  2's  from  20  and  from  19, 


49 

by  3's  from  21,  20,  and  19,  etc., — the  rhythmical  repe- 
tition resulting  in  the  readiest  memorizing  of  the  funda- 
mental differences,  however  they  may  have  been  derived. 
As  when  a  series  has  been  once  thoroughly  mastered,  it  is 
far  easier  to  learn  to  repeat  it  backwards,  than  it  was  to 
originally  master  it,  the  fundamental  differences  will 
be  more  readily  mastered  than  were  the  fundamental 
sums,  and  at  the  same  time,  the  fundamental  sums  will 
be  more  thoroughly  mastered  than  before.  Hence  the 
grouping  together  of  the  differences  obtained  by  the  suc- 
cessive subtraction  of  each  digit  first  from  its  multiple 
which  is  next  above  20  and  then  from  the  corresponding 
number  for  each  inferior  digit,  is  psychological: 

1.  For  those  pupils  who  failing  to  make  the  generali- 
zation find  in  the  mastery  of  the  effective  element  com- 
mon to  successive  groups  the  readiest  method  of  deriving 
the  fundamental  differences. 

2.  For  all  pupils  who  through  love  of  rhythm  find 
counting  the  form  of  repetition  by  which  they  can  most 
readily  memorize  the  differences, — whatever  the  method 
by  which  they  have  been  derived. 

And  this  grouping  is  psychological,  whether  the  funda- 
mental differences  are  taught  simultaneously  with  the 
fundamental  sums,  or  after  the  fundamental  sums  have 
been  thoroughly  mastered.24 

Granting  then,  that  a  priori  the  psychological  order  of 
teaching  the  fundamental  sums  and  differences  has  been 
satisfactorily  determined,  it  yet  remains  to  be  demon- 
strated that  these  a  priori  determinations  are  correct.  In 
the  schools  which  had  among  their  pupils  the  hundred 
children  whose  numerical  knowledge  was  tested  on  their 
first  entering  the  lowest  school  grade  this  demonstration 
has  been  attempted,23  and  the  results  of  six  months'  work 
subjected  to  a  rigid  and  impartial  test.  While  only  a 
wide  adoption  of  the  order  recommended,  in  schools  in 
which  the  conditions  greatly  vary,  will  make  possible  a 
series  of  general  tests,  which  will  be  universally  accepted 


50 

as  demonstrative, — the  results  so  far  attained  at  least 
serve  to  indicate  that  such  adoption  would  be  justifiable, 
and  that  such  experimental  demonstration  is  not  unlikely 
to  prove  the  determinations  to  be  indisputably  correct. 

Since  so  far  at  least  as  economical  derivation  is  con- 
cerned, the  results  of  teaching  in  psychological  order  are 
dependent  upon  the  employment  of  psychological  method, 
the  report  of  the  test  will  be  preceded  by  a  discussion  of 
method,  whose  determinations  like  those  concerning 
psychological  order,  the  results  of  the  test  will  tend  to 
confirm. 


CHAPTER  III.— THE  A  PRIORI  DETERMINA- 
TION OF  THE  PSYCHOLOGICAL  METHOD 
OF  TEACHING  THE  FUNDAMENTAL 
SUMS  AND  DIFFERENCES. 

I.  A  DEFINITION  OF  PSYCHOLOGICAL  METHOD. 

That  method  of  teaching  a  particular  branch  of  study 
is  psychological,  which  while  insuring  the  readiest 
mastery  of  its  subject-matter  by  the  individual  taught, 
gives  a  maximum  of  mental  training  along  the  lines  of 
development  for  the  furtherance  of  which  the  mastery  of 
that  subject-matter  is  peculiarly  adapted.  That  no  one 
method  will  be  psychological  for  all  individuals  must  of 
course  be  admitted.  Since,  however,  most  individuals 
are  taught  en  masse,  the  method  which  in  the  case  of  the 
great  majority  meets  the  conditions  just  specified,  is  the 
psychological  method  for  the  school.  In  the  absence  of 
any  satisfactory  demonstration  of  the  common  assump- 
tion that  all  special  mind  training  is  general  mind  training, 
it  is  safer  to  judge  what  training  should  result  from  the 
study  of  a  particular  branch,  both  in  the  light  of  that 
assumption,  and  in  that  of  its  negation.  Every  subject 
involves  in  its  mastery  the  predominance  of  some  particu- 
lar mental  activity.  All  other  activities  of  the  mind  are 
more  or  less  involved  in  it  as  they  are  more  or  less  in- 
volved in  every  mental  state,  but  one  predominates  in  the 
study  as  one  predominates  in  any  phenomenon  of  mind. 
Thus  imagination  is  the  ruling  activity  in  the  study  of 
descriptive  history  or  geography,  and  representation  in  the 
use  of  the  copy-book.  If  all  special  training  is  general 
training,  it  follows  that  so  far  as  the  school  curriculum  is 
concerned,  the  general  training — say  of  the  imagination — 
should  be  given  through  those  branches  in  whose  mastery 


52 

it  is  the  predominant  activity.  Otherwise  much  time  will 
be  wasted  in  overemphasizing  the  importance  of  some 
mental  activity,  which  is  only  incidentally  involved  in  the 
study  of  a  given  branch.  If  all  special  training  is  not 
general  training,  it  follows  that  the  training  of  the  imag- 
ination should  be  limited  so  far  as  it  is  not  purely  inci- 
dental, to  those  fields  of  activity  in  which  it  will  be 
naturally  exercised,  or  to  those  which  most  closely 
resemble  them.  In  either  case  the  method  of  teaching 
a  particular  branch  should  be  such  as  to  train  the  mental 
activity  necessary  to  its  mastery.  Other  activities  should 
be  incidentally  trained,  but  the  teaching,  in  addition  to 
mastery  of  subject-matter,  should  involve  training  con- 
centrated upon  the  activity  which  predominates  in  the 
science  or  art  which  is  being  taught.  Pupils  who  are 
mastering  descriptive  history,  should  reason  about  the 
facts  with  which  they  deal,  but  as  the  main  object  of  their 
study  is  to  make  those  facts  real,  the  method  of  instruc- 
tion should  be  planned  to  train  the  imagination,  rather 
than  the  reason.  In  other  words  the  intensive  study  of 
history  is  premature  wherever  it  precedes  or  is  taking  the 
place  of  the  realistic  presentation  of  important  events. 

II.  CONDITIONS    NECESSARY    TO    A    PSYCHOLOGICAL 

METHOD  OF  TEACHING  THE  FUNDAMENTAL 

FACTS  OF  NUMBER. 

The  important  place  that  mathematics  occupies  in  the 
curriculum  has  always  been  justified,  and  must  still  be 
justified,  if  it  is  to  be  justified  at  all,  by  the  fact  that  of  all 
the  so-called  mental  disciplines,  it  more  directly  and  more 
fully  than  any  other,  involves  the  pure  reasoning  activity 
of  the  human  mind.  If  the  special  training  in  pure 
reason,  popularly  supposed  to  be  involved  in  its  study,  is 
a  general  training  in  pure  reason,  it  is  the  duty  of  peda- 
gogy to  make  good  the  popular  belief.  If  not,  only  such 
mathematics  should  be  taught  as  can  be  permanently 


53 

mastered,  and  in  such  a  way  that  the  pupils  will  be  thor- 
oughly trained  in  such  mathematical  reasoning  as  they 
are  likely  to  exercise  in  everyday  life. 

In  either  case,  such  training  as  is  given  should  be 
mainly  a  training  of  reason ;  in  neither  should  it  be  mainly 
a  training  of  representation  or  imagination.  The  im- 
agination can  be  better  and  more  economically  trained  in 
some  study  in  which  it  is  the  dominant  mental  activity. 
The  visualizing  of  lines  and  angles,  and  plane  surfaces 
in  varying  perspective — all  the  trigonometrical  training 
recommended  by  Herbart  in  his  "A.  B.  C.  of  Sense  Per- 
ception/' may  or  may  not  be  a  necessary  part  of  the  train- 
ing of  the  child's  imagination.  It  certainly,  however,  is 
no  necessary  factor  in  the  study  of  number,  even  though  it 
be  a  preparation  for  the  visualizing  of  the  forms  of 
geometry.  Much  less  necessary  is  special  training  in  the 
visualizing  of  complex  columns  and  rows  of  figures.  In 
the  teaching  of  mathematics,  including  the  teaching  of 
the  fundamental  facts  of  number,  psychological  method  is 
that  which  while  insuring  the  readiest  mastery  of  the  facts 
by  the  individual,  will  give  a  maximum  of  training  to  the 
reasoning  activity  of  that  individual's  mind.  It  should 
assuredly  carry  with  it  training  of  other  mental  powers, 
but  just  as  assuredly  such  training  should  be  incidental  to 
the  training  of  reason. 

This  discussion  must  therefore  submit  method  to  the 
following  tests : 

1.  Is  a  given  method  better  adapted  than  any  other  to 
insure  the  readiest  derivation  by  the  great  majority  of 
the  pupils,  of  the  various  groups  of  fundamental  facts  ? 

2.  Is  it  better  adapted  than  any  other  to  insure  the 
readiest  memorizing  by  the  great  majority  of  the  pupils 
of  the  individual  facts  when  once  derived  ? 

3.  Does  it  involve  a  maximum  of  mental  training, 
chiefly  through  the  exercise  of  the  reasoning  activity? 

The  method  which  stands  these  tests  can  be  safely  char- 
acterized as  psychological. 


54 

III.  SPECIFICATION    OF   THE    GENERAL   METHODS    OF 
TEACHING  ELEMENTARY   NUMBER. 

However  ambiguous  the  term  "method"  may  be, — used 
now  by  Comenius  to  apply  to  his  general  analogy  between 
nature  and  instruction,  or  by  the  pedagogic  inventor  of 
to-day  to  designate  some  useful  but  restricted  device,  no 
one  will  deny  that  there  are  certain  general  methods  of 
instruction — call  them  what  you  will — based  upon  the 
various  ways  in  which  the  human  mind  acquires  knowl- 
edge. The  subject-matter  of  arithmetic  may  be  acquired 
inductively  or  deductively,  synthetically  or  analytically, 
concretely  or  abstractly.  It  has  been  shown  that  to  teach 
the  number-facts  in  psychological  order  is  for  the  most 
part  to  teach  inductively,  while  it  is  evident  that  to  so 
teach  the  fundamental  sums  is  to  teach  synthetically. 
That  this  teaching  is  here  psychological,  no  one  will  at- 
tempt to  disprove  who  remembers  that  it  has  for  its  object 
the  firm  association  in  the  mind  of  certain  number-facts, 
leaving  each  philosopher  to  his  own  devices  for  filling 
the  child  mind  with  his  number  "concept,"  though  being 
based  none  the  less  upon  the  fact  that  to  the  child  number 
is  an  expression  of  aggregation, — a  concept  with  which 
philosophers  are  beginning  to  agree.  While  it  is  per- 
haps equally  evident  that  the  psychological  order  of 
teaching  the  sums  and  differences  involves  a  method 
which  is  largely  abstract,  the  existing  prejudice  in  favor 
of  the  objective  method,  the  general  assumption  that  the 
concrete  basis  admittedly  necessary  to  the  comprehension 
of  the  abstract,  involves  the  objective  teaching  of  each  in- 
dividual fact  of  number,  makes  highly  desirable  a  thor- 
ough exposition  of  all  that  which  goes  to  prove  the 
abstract  method  psychological. 

With  this  end  in  view,  the  various  "methods"  of  deriv- 
ing and  memorizing  the  fundamental  facts,  may  all  be 
included  in  the  following  classification : 


55 

1.  The  Objective  Method,  in  which  each  individual 
fact  is  derived  by  the  direct  use  of  objects. 

2.  The  Memorial  Method  in  which  so  far  as  possible 
each  individual  fact  is  derived  by  the  summoning  up  of  the 
memorial  images  of  objects. 

3.  The  Abstract  Method  by  which  so  far  as  possible  the 
facts  of  each  group  are  derived  by  abstract  reason  from 
those  of  the  group  immediately  preceding. 

IV.  A  COMPARISON  OF  CERTAIN  OF  THESE  METHODS 

WITH  A  VIEW  TO  THE  DETERMINATION  OF 

PSYCHOLOGICAL  METHOD. 

(I)  THE  PESTALOZZIAN  OR  GRUBE  METHOD,  IN  WHICH 
EACH  INDIVIDUAL  FACT  is  DERIVED  WITH  THE 
DIRECT  AID  OF  OBJECTS. 

i.  Does  the  Grube  Method  Insure  the  Readiest  Deriva- 
tion of  the  Fundamental  Facts  of  Number? 

However  necessary  the  use  of  objects  to  the  formation 
of  the  number  "concept,"  however  essential  that  such  use 
should  result  in  the  discrimination,  abstraction  and  group- 
ing which  has  number  for  its  product,  it  by  no  means  fol- 
lows that  each  fact  of  number  should  be  objectively 
taught.25  In  order  that  an  individual  who  has  formed 
adequate  concepts  of  the  numbers  involved,  shall  fully 
comprehend  a  fundamental  sum  or  difference,  it  is  only 
necessary  that  he  shall  comprehend  the  simple  and  obvious 
fact  of  addition  or  subtraction,  and  perceive  for  himself 
the  invariable  result.  The  objective  derivation  of  indi- 
vidual facts  is  justifiable,  if  in  order  to  perceive  that  the 
result  is  invariable,  in  order  to  intelligently  accept  it  as 
an  abstract  fact,  it  must  first  be  perceived  to  be  true  in 
various  concrete  forms,  and  then  the  generalization  made 
that  being  true  of  these,  it  is  true  of  all, — that  3  cubes 
and  2  cubes  being  5  cubes,  3  splints  and  2  splints,  5  splints, 


56 

etc.,  3  and  2  must  invariably  be  5.  It  is  a  serious  mistake 
however,  to  assume  that  it  is  necessary  to  repeat  this 
mental  process  in  deriving  every  fundamental  sum  or 
difference.  Sooner  or  later,  the  generalization  is  made 
that  every  fact  of  number,  however  derived,  is  universal 
in  its  application,  that  if  2  and  i  are  invariably  3,  and 
3  and  2  invariably  5,  3  and  4,  found  to  be  7  in  one  in- 
stance, whether  by  a  derivation  that  is  concrete  or  one  that 
is  abstract, — will  be  known  to  be  7,  no  matter  what  the 
concrete  form  of  the  addition  may  be.  No  one  would 
dream  of  teaching  36  and  2  objectively,  in  order  to  lead 
pupils  to  perceive  by  the  repeated  addition  of  various 
objects  that  whatever  things  may  be  added,  36  and  2  will 
always  equal  38.  They  already  know  that  if  through 
abstract  addition,  36  and  2  be  found  to  equal  38,  the  result 
will  be  invariable,  whether  the  objects  concerned  are 
splints  or  trilobites.  It  remains  for  advocates  of  concrete 
number  derivation  to  show  that  this  is  not  equally  true 
of  6  and  2.  Generalization  requires  but  few  particulars. 
After  the  child  has  found  that  2  and  i — 3  with  one  set  of 
objects,  are  still  3  though  the  objects  be  varied, — and 
being  3  though  the  objects  be  varied,  are  therefore  3 
whatever  the  objects  may  be,  he  will  not  require 
many  repetitions  of  this  process  of  abstraction  and  gen- 
eralization in  the  derivation  of  other  facts,  before  he  will 
assume  that  each  fact  which  he  derives  will  likewise  apply 
to  all  things.  In  cases  where  the  abstract  fact  seems  to 
be  derived  by  the  summoning  up  of  one  that  is  concrete, 
— as,  for  example,  where  a  pupil  failing  to  give  the  ab- 
stract 3  and  5  can  be  led  to  recall  the  fact  that  3  marbles 
and  5  marbles  are  8  marbles,  would  seem  to  indicate  that 
abstraction  and  generalization  had  not  yet  taken  place,  if 
it  were  not  that  the  recall  is  often  followed  by  the  im- 
mediate generalization — 3  and  5  are  8.  It  is  not  that 
3  and  5  had  not  been  generalized,  but  that  not  having 
been  thoroughly  memorized,  the  familiar  concrete  associa- 
tion helped  in  its  recall. 


57 

As  then,  the  use  of  objects  is  not  necessary  to  intelli- 
gent generalization  in  the  case  of  each  fundamental  fact, 
it  having  already  been  clearly  shown  in  the  discussion  of 
psychological  order,  that  the  fundamental  sums  and  dif- 
ferences can  be  more  readily  derived  by  a  simple  pro- 
cess of  abstract  reason,  it  necessarily  follows  that 
objects  should  be  used  only  in  general  drill  in  num- 
ber perception  and  in  the  occasional  verification  of  facts  al- 
ready derived,  except  in  so  far  as  their  use  is  necessary 
to  the  mastery  of  the  effective  element  common  to  each 
group.26  The  Grube  method,  therefore,  objectively  de- 
riving each  individual  fact,  would  seem  at  least  as  regards 
derivation,  to  be  uneconomical  and  unpsychological. 

2.  Does  it  Insure  the  Readiest  Memorising  of  the  Facts 
When  Once  Derived? 

When  a  number-fact  has  been  once  derived,  it  must  be 
memorized  by  repetition, — now  as  in  the  days  of  Richter, 
and  as  it  must  ever  be,  "the  mother  not  only  of  study  but 
also  of  education."  Just  so  far  as  the  Grube  method 
limits  its  memory  drill  to  repetition  with  actual  objects  or 
of  concrete  facts,  it  is  unpsychological.  There  is  a  ten- 
dency among  the  friends  of  the  "new  education"  to  con- 
fuse the  mechanical  with  the  irrational.  Word  for  word 
memorizing  of  the  subject-matter  of  education  is  mechani- 
cal, but  it  is  irrational  not  on  account  of  its  being  thus 
mechanical,  but  first,  because  the  words  of  few  text-books 
are  worth  memorizing,  and  second,  because  thoughts  are 
more  certain  of  recall  through  their  manifold  associa- 
tion with  other  thoughts.  If  words  are  to  be  thoroughly 
memorized, — no  matter  how  full  the  apperception  of  the 
thoughts  which  they  express,  the  necessary  repetition 
should  be  "mechanical."  So  with  the  abstract  fact  of 
number.  Five  is  to  be  permanently  associated  with  3 
and  2, — an  association  of  symbols  and  not  of  thoughts. 
This  association  is  assisted  by  the  common  association  of 


58 

the  5  and  the  3  and  2  with  cubes,  splints  and  geometrical 
forms,  only  through  the  repetition  involved  in  the  process. 
At  least  no  individual  is  likely  to  recall  a  number- fact 
because  of  its  association  with  the  same  series  of  objects 
with  which  all  other  number- facts  have  been  associated. 
But  such  repetition  is  plainly  uneconomical  when  com- 
pared with  the  mechanical  repetition  of  the  abstract  fact, 
unless,  to  be  sure,  the  former  is,  as  is  assumed,  more  in- 
teresting to  the  child,  and  so  holds  his  attention  as  the 
latter  can  not.  It  is,  however,  at  least  very  questionable, 
whether  this  assumption  of  interest  will  stand  the  test  of 
impartial  investigation.  The  use  of  a  great  variety  of 
attractive  objects  interests  the  child,  but  in  the  objects, 
not  in  the  fact  which  they  are  intended  to  exemplify, 
while  the  use  of  the  same  set  of  objects  again  and  again, 
— no  matter  how  finely  polished  the  cubes  or  how  brightly 
colored  the  splints, — is  no  stimulus  to  interest,  especially 
as  in  the  end  it  must  come  to  be  associated  with  failure 
to  recall  and  sometimes  with  more  or  less  merited  rebuke. 
Such  interest  as  is  manifested  in  concrete  repetition  may 
after  all  be  traceable  to  a  love  of  counting  combined  with 
the  joy  of  mastering  a  new  fact  or  recalling  one  that  was 
forgotten,  supplemented  in  many  cases  by  the  sympathy, 
enthusiasm  and  dramatic  ability  of  a  Pestalozzian 
teacher,  rather  than  by  the  inert  concreteness  of  Festaloz- 
zian  matter. 

There  is  possible,  however,  another  form  of  concrete 
repetition  in  which  a  given  number-fact  is  firmly  associ- 
ated with  a  particular  concrete  thing,  which  often  seen, 
will  repeatedly  recall  the  fact  to  mind.  2  +  2  are  4  may 
thus  be  associated  with  the  legs  of  a  chair,  or  2  and  i  are  3 
with  the  vertices  of  a  triangle.  While  granting  that  the 
numerical  association  would  often  arise  when  the  objects 
were  seen,  and  that  this  incidental  and  natural  repetition 
would  firmly  fix  the  facts  in  mind,  it  would  be  mani- 
festly impossible  to  select  a  sufficient  number  of  familiar 
objects  adapted  to  this  system  of  repetition. 


59 

A  priori,  then  it  would  seem  that  any  form  of  abstract 
repetition  of  a  number-fact  once  it  has  been  intelligently 
derived,  so  limited  in  time  as  not  to  become  wearisome, 
will  result  in  a  readier  memorizing  than  will  repetition 
in  the  concrete.  Especially  will  this  be  the  case,  where 
the  repetition  can  be  made  rhythmical  and,  so,  pleasing  to 
the  child. 

The  Grube  method  therefore  can  not  be  psychological 
in  so  far  as  it  involves  repeated  objective  illustration  in 
the  memorizing  of  fundamental  facts.  Its  use  of 
memorial  images  or  of  concrete  associations  to  the  same 
end,  will  be  considered  later  on  in  this  discussion. 

3.  Does  it  Involve  a  Maximum  of  Mental  Training, 
ChieHy  Through  the  Exercise  of  the  Reasoning  Activity? 

The  most  enthusiastic  advocates  of  objective  operation 
are  hardly  likely  to  urge  it  as  a  means  to  mental  training. 
They  rather  assume  it  as  a  necessary  condition.  The 
discrimination  into  separate  wholes  of  objects  so  similar 
that  they  can  be  perceived  as  a  united  whole,  however 
necessary  to  "number  perception/'  is  not  a  mode  of  train- 
ing necessary  to  supplement  the  child's  experience.  If 
the  similarity  were  to  be  sought  out,  it  would  be  valuable 
training  of  an  essential  factor  of  the  reasoning  process, 
but  this  would  be  merely  adding  difficulty  to  the  formation 
of  the  "number  concept."  This  analysis  of  wholes  into 
component  wholes  easily  apparent,  far  from  training 
sense  perception,  serves  but  to  strengthen  the  every-day 
experience,  which  discriminates  wholes  without  perceiv- 
ing them  in  detail.  But  even  were  training  in  sense  per- 
ception and  in  the  discovery  of  concrete  similarities  in- 
volved in  the  teaching  of  the  fundamental  sums  and  differ- 
ences, it  should  be  purely  incidental  to  the  training  of 
abstract  reason.  Only  in  systematic  nature-study  and 
the  experimental  study  of  the  natural  sciences  should 
such  training  become  a  dominant  factor. 


6o 

The  only  remaining  form  of  mental  training-  involved 
in  the  objective  method  is  exhaustive  drill  in  generaliza- 
tion. It  has  already  been  demonstrated  that  continual 
repetition  of  this  process  is  unnecessary  in  so  far  as  it  is 
intended  to  lead  the  child  to  comprehend  that  a  fact  true 
for  one  collection  of  objects  will  hold  for  all.  Generali- 
zation, and,  too  often,  generalization  from  insufficient 
particulars,  is  a  natural  activity  of  the  human  mind.  3 
and  2 — 5  in  the  one  case  actually  tested — will  be  5  in  all, 
unless  it  is  proven  to  be  6.  It  is  only  so  far  as.  the  ob- 
jective method  acts  as  a  check  upon  this  dangerous  ten- 
dency that  it  can  be  useful  as  training  in  generalization. 
As  the  fact  that  3  and  2  will  always  prove  to  be  5  only 
serves  to  confirm  the  usual  assumption,  training  which 
looks  toward  a  surer  judgment  should  be  reserved  for 
studies  which  demonstrate  that  one  Emerald  Island  does 
not  make  all  islands  green,  or  that  a  broken  rule  does  not 
infallibly  indicate  a  malicious  child. 

Except  in  so  far  as  all  mental  activities  are  involved 
in  each,  this  is  the  only  mental  training  involved  in  the 
Grube  method.  It  not  only  fails  to  train  the  abstract 
reason,  but  involves  the  very  minimum  of  mental  training 
which  can  result  in  the  mastery  of  the  fundamental  sums 
and  differences. 

(II)  THE  MEMORIAL  METHOD,  IN  WHICH,  so  FAR  AS 
POSSIBLE,  EACH  INDIVIDUAL  FACT  is  DERIVED  AND 
MEMORIZED  BY  THE  SUMMONING  UP  OF  MEMORIAL 
IMAGES  OR  OF  CONCRETE  ASSOCIATIONS. 

i.  Does  the  Memorial  Method  Insure  the  Readiest 
Derivation  of  the  Fundamental  Number-Facts? 

All  that  has  been  urged  against  the  excessive  use  of 
objects  in  teaching  the  fundamental  facts  of  number 
can  be  urged  with  more  than  equal  force  against  such 
teaching  as  is  dependent  upon  the  summoning  up  of 


6i 

memorial  images.  Indeed  no  one  would  seriously  advo- 
cate the  deriving  of  7  as  3  +  4  through  the  recall  of  the 
visual  images  of  3  objects  and  4  objects  and  their  com- 
bination into  7.  The  memorial  method,  however,  has 
been  very  generally  used  in  the  memorizing  of  facts  when 
once  they  have  been  objectively  derived,  or  objectively  ap- 
plied, whether  as  the  result  of  instruction  or  of  numerical 
experience  outside  of  the  school. 

2.  Does  it  Insure  the  Readiest  Memorizing  of  the 
Facts  When  Once  Derived? 

It  has  already  been  shown  that  the  very  general  use  of 
"number  stories" — however  valuable  as  a  drill  in  lan- 
guage,— is  unnecessary  as  a  drill  in  the  application  of  a 
generalized  fact.  If  the  child  knows  that  4  and  2  are  6, 
he  knows  that  4  oranges  and  2  oranges  are  6  oranges 
without  ever  having  applied  that  particular  abstract  sum 
to  the  limited  stock  of  objects  which  he  has  come  to  use 
in  his  number  stories.  Unnecessary  so  far  as  the  sums 
and  differences  are  concerned,  as  a  means  to  application, 
the  story  even  more  than  the  use  of  actual  objects,  is 
uneconomical  as  a  mode  of  repetition.  If  the  child  really 
visualizes  as  numbered  groups  the  objects  that  he  names — 
which  will  presently  be  disputed, — all  that  has  been  urged 
against  objective  repetition,  is  fully  applicable,  with  the 
additional  argument  that  the  visualizing  is  unnecessary. 
If  he  merely  visualizes  unnumbered  objects  or  associates 
the  unfamiliar  fact  with  the  name  of  some  familiar  con- 
crete thing,  his  repetition  is  plainly  uneconomical.  The 
argument  that  he  will  be  more  interested  in  the  concrete 
has  been  already  met.  That  the  association  of  the  un- 
familiar with  the  familiar  will  result  in  its  recall  and  so 
in  its  frequent  repetition  has  also  been  shown  to  be 
improbable.  It  only  remains  to  be  demonstrated  that 
even  though  it  is  assumed  that  the  child  visualizes  the 
objects  used  as  addends  or  as  subtrahend  and  difference, 


62 

he  can  not  visualize  sum  or  minuend  as  a  numbered 
whole. 

It  is  a  generally  accepted  fact  that  but  five  or  six  objects 
can  be  simultaneously  perceived,  if  indeed  it  has  been 
satisfactorily  demonstrated  that  even  this  number  can  be 
perceived  simultaneously.  It  is  possible  through  training 
to  perceive  more  than  five  or  six  objects  instantaneously, 
but  in  all  cases  where  this  has  been  successfully  accom- 
plished, it  seems  to  be  as  the  result  not  only  of  a  well- 
trained  observation,  but  of  an  estimation  so  ready  as  in 
many  cases  to  be  quite  unconscious.  That  is,  the  percep- 
tion is  in  all  probability  successive  and  not  simultaneous. 
Where  the  objects  to  be  perceived  have  been  so  placed  as 
to  prevent  so  far  as  possible  their  separation  into  more 
or  less  definite  groups,  patient  and  persistent  training  has 
failed  to  increase  the  number  of  objects  "simultaneously" 
perceivable.27  Hence  it  is  obvious  that  the  individual, 
looking  at  the  group  of  objects  which  for  the  first  time  he 
has  formed  by  the  addition  of  say  8  to  9,  can  have  no 
numerical  concept  directly  arising  from  his  perception 
of  the  group  as  a  whole.  It  is  not  until  he  perceives  it  as 
four  4's  and  i,  three  5's  and  2,  or  some  other  combination 
with  which  he  is  familiar,  or  until  he  has  counted  the 
objects  composing  it,  that  he  knows  it  to  be  17.  That  is, 
he  infers  its  equality  with  the  group  resulting  from  the 
addition  of  two  other  groups,  by  perceiving  that  it  may 
be  exactly  separated  into  those  groups,  or  by  counting 
proves  it  to  be  equal  to  16  and  i.  The  only  visual  image, 
then,  that  can  arise  in  his  mind  in  response  to  the  oral  or 
visual  "17,"  is  not  that  of  one  group  containing  seventeen 
objects,  but  of  a  group  composed  of  at  least  three  other 
groups — as  6  and  6  and  5. 

While  it  has  yet  to  be  determined  to  what  percentage 
of  those  versed  in  elementary  number  such  a  visual  image 
does  actually  present  itself,  it  is  evident  that  in  the  case 
of  larger  numbers,  the  increasing  complexity  of  an  image 
— no  group  of  which  can  be  greater  than  6, — precludes 


63 

its  presentation  when  the  number  which  it  would  visually 
represent  is  seen  or  named.  Even,  for  example,  so  small 
a  number  as  57  could  not  be  visually  represented  by  fewer 
than  nine  groups  of  6  each  and  one  of  3.  It  is  perhaps 
safe  to  say  that  under  ordinary  conditions,  in  the  case  of 
the  great  majority  of  individuals,  no  visual,  tactile,  audi- 
tory, or  kinsesthetic  image  of  a  group  of  objects  or  sensa- 
tions, rises  in  response  to  a  number-name.  And  it  may 
be  asserted  with  still  more  certainty  that  no  such  repro- 
duction is  necessary  to  the  number  concept.28 

There  remains  for  consideration  that  form  of  the  me- 
morial method  in  which  the  concrete  association  is  with 
the  written  number  symbol.  Figures  are  objects.  Should 
the  abstract  fact  after  it  has  been  intelligently  derived 
be  memorized  through  the  repetition  of  its  written  form  ? 
So  far,  the  discussion  has  concerned  itself  with  the  oral 
memorizing  of  the  abstract  fact,  the  desirability  for  which 
no  one  is  likely  to  deny.  But  much  of  the  addition  and 
subtraction  which  the  majority  of  individuals  are  called 
upon  to  perform  is  not  oral  but  written.  More  than  this, 
there  are  those  who  favor  such  drill  in  the  visualizing  of 
the  written  process,  that  long  and  difficult  problems  can 
be  solved  mentally.  While  the  memorizing  of  the  oral 
fact  is  all  that  is  necessary  to  written  addition  or  written 
subtraction,  while  either  operation  may  be  made  a  purely 
oral  one  by  the  oral  repetition  of  the  numbers  written, — 
it  is  obvious  that  operation  will  be  more  rapid  where  a 
visual  or  a  semi-visual  association  has  been  formed.  In 
written  subtraction  the  sight  of  the  digits  to  be  subtracted 
should  at  once  call  up  the  oral  or  the  visual  differences, 
without  the  intermediation  of  the  oral  digit  names, — a 
fact  that  is  equally  true  of  addition  not  in  column.  To 
this  end,  then,  concrete  association  is  necessary.  The 
pupil  should  be  drilled  upon  saying  or  writing  the  result, 
when  the  numbers  operated  upon  are  written  one  above 
the  other, — it  mattering  not  whether  the  necessary  kin- 
sesthetic chain  is  recalled  by  its  visual  or  its  auditory 


64 

image.  The  algebraic  form, — i  +  3  =  4  Or  5  —  2  =  3, 
so  common  in  the  written  work  of  the  school  is  no  help 
to  this  written  association.  It  is  only  justifiable  in  so  far 
as  it  is  desirable  to  familiarize  the  pupils  with  algebraic 
expression,  and  has  in  all  probability  resulted  not  from 
such  justification,  but  as  a  survival  of  the  old  addition  and 
subtraction  tables  scattered  by  the  Grube  method  into  indi- 
vidual facts. 

A  different  form  of  concrete  association — the  "semi- 
visual" — is  necessary  to  the  highest  degree  of  facility  in 
addition  in  column.  After  the  addition  of  the  two  digits 
at  the  top  of  the  column,  the  numbers  to  be  added  are  not 
seen  together,  nor  are  they  successively  pronounced.  The 
addition  is  not  of  two  numbers  written  one  above  the 
other,  but  of  a  spoken  number  or  its  auditory  image,  with 
a  number  written  in  column, — each  oral  result  or  its 
auditory  image  being  added  to  the  number  next  above  or 
below  the  last  number  added.  It  is  evident  that  to  insure 
rapid  addition  in  column,  drill  on  neither  the  oral  nor  the 
written  abstract  facts  alone,  is  all  that  is  needed.  The 
drill  should  consist  of  the  addition  of  a  written  digit 
to  one  that  has  been  spoken. 

The  memorial  method,  then,  has  to  be  utilized  in  the 
drill  preliminary  to  rapid  operation. 

3.  Does  it  Involve  a  Maximum  of  Mental  Training 
Chiefly  Through  the  Exercise  of  the  Reasoning  Activity? 

It  is  at  once  apparent  that  teaching  of  number  by  the 
memorial  method  involves  more  mental  training  than  does 
that  which  is  limited  to  the  objective  method  alone.  Here 
the  power  of  the  mind  to  recall  the  concrete,  especially  the 
visualizing  power,  is  more  or  less  exercised.  But 
whether  or  not  such  training  is  general  training,  but  little 
of  it  is  necessary  in  the  teaching  of  number,  while  it  can 
be  made  far  more  effective  in  the  study  of  drawing,  geo- 
graphy or  history  in  which  it  is  an  essential  factor.  More 


65 

than  likely  the  power  to  visualize  unnumbered  objects  and 
geometrical  forms  is  all  the  visualizing  power  desirable 
in  mathematics.  Whether  this  power  is  developed  by  the 
memorial  method  is  to  say  the  least  very  doubtful.  The 
representation  which  it  usually  involves  is  in  all  likeli- 
hood, that  of  vague  and  indistinct  wholes,  only  resulting 
in  images  clear  and  distinct  in  all  their  parts  when  visual- 
izing is  made  a  main  end  of  the  number  study,  as  in  the 
case  of  the  mental  solution  of  lengthy  problems.  It  is 
very  questionable  whether  this  power  to  visualize  a  com- 
plex numerical  operation  is  a  useful  accomplishment. 
Occasions  are  rare  when  ready  written  operation  will  not 
fully  serve  the  same  purpose.  It  is  necessary  to  discrimi- 
nate sharply  between  fundamental  number  fully  compre- 
hended, and  other  distinct  ends  with  which  it  has  been 
confounded.  Even  correlation  does  not  demand  that 
the  economical  teaching  of  one  branch  shall  be  sacrificed, 
and  the  main  end  of  the  consequent  training  minified,  in 
order  that  something  else  more  or  less  related  to  its  sub- 
ject-matter or  more  or  less  involved  in  the  training  shall 
be  taught  with  it. 

(Ill)  THE  ABSTRACT  METHOD,  IN  WHICH,  so  FAR  AS 
POSSIBLE,  EACH  OF  THE  FACTS  IN  EACH  GROUP 
ARE  DERIVED  BY  ABSTRACT  REASON  FROM  THOSE 
OF  THE  GROUP  IMMEDIATELY  PRECEDING,  AND 
LARGELY  MEMORIZED  BY  ABSTRACT  COUNTING. 

i.  Conclusion  that  the  Abstract  Method  Insures  the 
Readiest  Derivation  of  the  Fundamental  Number-Facts. 

As  it  is  only  through  the  use  of  this  method  that  the 
psychological  order  of  teaching  number  can  be  fully  util- 
ized in  the  derivation  of  the  fundamental  sums  and  dif- 
ferences, it  is  plainly  the  psychological  method,  wherever 
it  is  adapted  to  the  majority  of  the  children  taught. 
Objects  are  used  only  in  so  far  as  they  are  necessary  to  the 


66 

comprehension  of  the  effective  element  common  to  each 
group,  to  the  occasional  verification  of  a  fact  abstractly 
derived,  and  to  general  drill  in  number  perception.  The 
intelligent  derivation  of  one  fact  in  a  group  insures  the 
ready  derivation  of  all.  Indeed  with  many  pupils,  the 
intelligent  derivation  of  the  facts  in  the  first  two  or  three 
groups  insures  the  ready  derivation  of  all  the  remaining 
facts.  By  no  other  method  can  the  fundamental  facts 
be  so  readily  derived. 

2.  Conclusion  That  It  Insures  the  Readiest  Memoriz- 
ing of  the  Facts  When  Once  Derived. 

That  the  memorizing  of  the  fundamental  sums  and  dif- 
ferences by  forward  and  backward  counting  by  2's,  3's, 
4's,  etc.,  is  economical,  has  been  plainly  demonstrated. 
It  will  become  unpsychological  only,  if  as  the  series 
multiply,  the  child  who  enjoys  counting  by  I's  and  by  2's 
should  become  weary  of  counting  by  3's,  4's  and  5's.  This 
constitutes  the  main  method  of  oral  drill.  It  should  of 
course  be  supplemented  by  the  abstract  written  drill  in 
which  with  the  two  addends  or  the  minuend  and  subtra- 
hend written,  the  pupil  calls  out  or  writes  the  proper 
result,  and  the  semi-visual  drill  preliminary  to  addition 
in  column,  in  which  the  pupil  gives  the  sum  of  the  number 
named  by  the  teacher  and  that  indicated  by  a  pointer  in 
a  column  containing  all  the  digits. 

By  means  of  these  drills,  not  only  may  the  facts  be 
readily  and  economically  memorized,  but  if  for  the  first 
year  or  so,  they  are  made  preliminary  to  all  working  of 
examples  there  need  never  be  any  slow  operation.  From 
the  first,  pupils  should  not  be  allowed  to  engage  in  opera- 
tion involving  facts  which  they  can  not  give  with  the 
highest  degree  of  facility.  Examples  should  be  devised 
made  up  wholly  of  those  sums  or  differences  upon  which 
thorough  drill  has  been  given  immediately  before  the 
addition  or  the  subtraction  is  to  be  performed.  Thus 


67 

operation  being  invariably  rapid,  rapid  operation  will  be- 
come a  habit — the  degree  of  rapidity  being  dependent 
upon  the  reaction  time  of  the  individual,  and  finding  its 
maximum  at  the  point  where  rapidity  threatens  to  result 
in  nervousness.29 

3.  Conclusion  That  It  Involves  a  Maximum  of  Mental 
Training — Chiefly  in  the  Exercise  of  the  Reasoning  Ac- 
tivity. 

The  abstract  method  not  only  makes  possible  the  readi- 
est derivation  and  the  most  economical  memorizing,  but 
involves  a  maximum  of  mental  training — chiefly  in  the 
exercise  of  the  reasoning  activity.  If  adapted  to  the 
individual  taught,  it  is  therefore  not  only  the  mathemati- 
cal, but  unquestionably  the  psychological  method  of 
teaching  the  fundamental  sums  and  differences. 


CHAPTER  IV.— RESULTS  OF  A  HALF  YEAR'S 
TEACHING  IN  THE  ORDER  AND  ACCORD- 
ING TO  THE  METHOD  JUST 
DETERMINED. 

I.  CONDITIONS    UNDER    WHICH    THE    ATTEMPT    WAS 

MADE. 

These  a  priori  determinations  were  put  to  the  test  in  the 
same  schools  in  which  effort  had  been  made  to  ascertain 
the  numerical  knowledge  possessed  by  the  child  on  enter- 
ing school.  On  the  whole,  existing  conditions  were  dis- 
tinctly unfavorable  to  success.  In  two  of  the  schools  the 
teachers  were  new  and  with  practically  no  experience  in 
teaching  young  children,  while  a  third,  after  being  taught 
for  about  three  weeks  by  a  teacher  in  the  early  stages 
of  typhoid  fever,  and  for  two  months  by  a  substitute,  was 
then  placed  in  charge  of  a  young  girl,  well-trained,  but 
who  had  never  taught  before.  The  number  of  pupils 
under  each  teacher  varied  from  twenty-eight  to  fifty- four, 
and  averaged  forty-five.  In  two  out  of  the  seven  rooms, 
the  great  majority  of  pupils  were  beginners;  in  the 
remainder,  most  of  the  children  had  entered  during  the 
last  school  year,  while  many  had  been  in  school  for  a  still 
longer  period.  No  special  attention  was  given  to  begin- 
ners. On  the  contrary,  according  to  a  long-established, 
though  unfortunate  custom,  far  less  time  was  devoted  to 
those  classes  which  largely  consisted  of  children  just 
entering  school,  than  to  those  in  which  the  great  majority 
were  hold-overs.  More  than  this, — the  same  amount  of 
time  was  not  allowed  to  corresponding  classes  in  the  dif- 
ferent schools,  one  class  of  fourteen,  for  example,  being 
given  on  an  average  twenty  minutes  of  oral  drill  each  day, 
while  another  of  twelve  was  given  twenty  minutes  of  simi- 
lar instruction.  The  more  advanced  of  the  beginners 
were  not  taught  with  others  who  had  just  entered,  but  in 

(68) 


69 

classes  in  which  hold-overs  largely  predominated.  As  the 
schools  did  not  open  until  September  5,  and  the  classes 
were  tested  during  the  first  two  weeks  in  March,  the 
results  on  the  average  represent  the  work  of  about  one 
hundred  and  twenty  days — which  for  half  of  those  first 
entering  school  could  not  have  meant  an  approximate 
total  of  thirty  hours  of  actual  instruction  and  for  the 
remainder  from  forty  to  fifty  hours.  As  it  was  believed 
that  the  experience  of  teachers  during  a  short  period, 
unrestricted  by  a  priori  suggestions  as  to  method  of  pro- 
cedure, would  be  the  only  stable  basis  for  detailed  direc- 
tions in  the  future,  they  were  given  no  special  instructions 
other  than  that  they  should  take  up  the  elementary  facts 
in  the  order  just  described,  and  in  such  a  way  that  the 
pupils  should  so  far  as  possible  be  led  to  perceive  each 
common  element. 

In  view  of  the  variability  in  teaching  power,  in  length 
of  drill  period  and  in  manner  of  teaching,  taken  in  con- 
nection with  the  inexperience  of  some  of  the  teachers,  the 
few  hours  spent  in  actual  instruction  in  number,  the 
absence  of  detailed  directions,  and  finally  the  fact  that 
even  the  few  and  simple  directions  that  were  given  were 
for  a  time  in  some  cases  misunderstood, — the  highly  satis- 
factory progress  of  the  work,  from  the  very  first,  appears 
to  justify  the  a  priori  determination  of  psychological 
order  and  method. 

Before  proceeding  to  discuss  the  searching  and  im- 
partial test  given  at  the  close  of  the  first  half  year,  with 
its  significant  though  of  course  inconclusive  results,  it 
will  be  profitable  to  consider  the  suggestive  data  inci- 
dentally furnished  by  the  work  that  was  subjected  to  it. 

II.  SOME  OBSERVATIONS  UPON  FACTS  WHICH  IT  INCI- 
DENTALLY FURNISHED. 

While  no  children  were  reported  as  having  any  serious 
difficulty  with  mechanical  counting,  it  was  soon  dis- 
covered that  probably  from  10  per  cent  to  20  per  cent  of 


yo 

the  whole  number  of  beginners  would  find  it  hard  to 
master  the  element  common  to  the  facts  of  even  the  first 
group — although  such  mastery  was  necessary  to  intelli- 
gent counting.30  The  same  children  almost  without  ex- 
ception also  had  trouble  in  their  word-study, — in  the  asso- 
ciation of  the  word  and  its  symbol.  Indeed  at  this  point 
it  may  be  well  to  state  that  among  the  children  tested,  it 
was  a  very  exceptional  thing  to  find  those  who  were  in 
more  advanced  reading  classes  than  number-classes  or 
vice  versa, — only  three  being  classified  higher  in  number 
and  two  in  reading.  The  "slowness,"  then,  of  these 
beginners,  who  with  some  "backward"  hold-overs  from 
the  year  before,  composed  the  lowest  number  section  of 
each  first  grade  school,  was  plainly  due  to  some  condition 
common  to  both  number-work  and  word-study, — a  con- 
dition which  could  be  no  other  than  the  complexity  of  the 
association  required  to  memorize  word-symbol  or  number- 
fact.  Children  whose  associations  had  so  far  been  largely 
confined  to  those  uniting  a  name  with  a  thing  or  one 
name  with  another,  found  it  well  nigh  impossible  to  asso- 
ciate a  name  with  a  complex  written  symbol,  or  the  "3," 
"and,"  "2,"  "are,"  and  "5,"  into  a  fundamental  sum.  As 
they  could  readily  associate  a  single  letter  with  the  cor- 
responding sound,  and  a  number  with  that  adjacent  to  it 
in  the  number  scale,  the  mechanical  counting  to  10  or  20 
already  begun  on  their  entering  school  was  readily  mas- 
tered. But  how  were  they  to  be  led  from  that  as  a  point 
of  departure,  to  the  derivation  and  memorizing  of  the 
essential  facts  ?  All  that  was  necessary  to  the  derivation 
of  the  first  group  was  their  comprehension  of  the  fact  that 
the  addition  of  one  to  a  number,  results  in  the  number 
next  above  it  in  the  scale.  But  at  the  start,  the  pupils 
comprising  these  lower  classes,  utterly  failed  to  compre- 
hend it.  Hence  the  effort  was  made  to  simplify  the 
process  of  instruction.  After  the  children  could  mechani- 
cally count  with  a  high  degree  of  readiness  from  i  to  9, 
the  teacher  began  to  interject  the  expression  "and  one," 


between  the  terms  of  the  series  as  given  by  the  pupils  in 
counting  the  objects  as  she  added  them  one  by  one.  Thus 
after  they  had  become  accustomed  to  the  interruption,  the 
"and  one"  gradually  came  to  carry  with  it  its  proper 
meaning,  and  to  be  firmly  associated  with  each  number 
and  that  which  immediately  followed  it.  The  second 
step  in  the  process  was  to  drill  them  upon  the  ready  giving 
of  the  number  that  came  next  in  the  scale  after  any  other 
which  the  teacher  called  out.  Finally  they  were  led  to 
observe  that  the  addition  of  one  denoted  by  "and  one" 
always  resulted  in  that  now  familiar  "next"  number. 
Within  six  weeks  all  but  six  of  the  "slow"  ones  had 
mastered  the  facts  of  this  group,  together  with  the  inver- 
sions. Under  the  old  plan  they  would  have  been  vainly 
struggling  to  remember  the  two  or  three  "simpler"  facts 
which  they  had  counted  out  with  the  objects  every  day. 
During  the  first  half  year,  even  the  remaining  six  yielded 
to  treatment  and  mastered  the  nineteen  facts  with  their 
inversions.  A  little  less  than  one- fourth  of  the  beginners, 
however,  in  the  corresponding  time,  failed  to  associate 
20  with  two  IQ'S,  30  with  three  ID'S,  etc.  As  they 
were  led  to  think  of  the  "ty"  as  a  short  way  of  saying 
"tens,"  and  permitted  to  say  "twoty,"  "threety,"  and 
"fivety,"  in  all  cases  where  the  corresponding  contractions 
did  not  seem  to  be  sufficiently  suggestive,  and  as  those 
classes  who  to  the  last  pupil  mastered  these  facts  with  but 
little  difficulty,  were  just  as  "slow"  as  those  who  to  the 
last  pupil  failed,  the  true  reason  for  their  failure  to  master 
this  and  the  two  following  groups  appears  to  lie  not  so 
much  in  their  inability  to  learn,  as  in  the  false  assump- 
tion on  the  part  of  their  teachers  that  having  had  difficulty 
with  the  numbers  from  i  to  10,  there  was  little  use  in 
attempting  to  teach  them  the  "larger"  and  "more  diffi- 
cult" numbers  from  20  to  zoo.31 

While  at  the  close  of  the  first  half  year  34  per  cent  of 
the  beginners  had  not  yet  succeeded  in  firmly  memorizing 
all  the  fundamental  sums  resulting  from  the  addition  of 


72 

2)  very  few  of  them  had  failed  to  comprehend,  that  the 
addition  of  2  was  equivalent  to  the  addition  of  i  and  i, 
Great  care,  however,  had  to  be  taken  to  avoid  plunging 
those  who  were  more  immature  into  inextricable  mental 
confusion — the  result  being  brought  about  rather  through 
suggestion  than  by  syllogism.  For  example,  the  series 
"4,  6,  8,"  etc.,  was  taught  as  follows:  The  teacher  plac- 
ing two  cubes  in  plain  view  of  the  class, — "How  many 
cubes  have  I  ?"  adding  one  and  then  another — "Two  and 
i  ?  and  i  ?"  putting  the  two  new  blocks  together — "What 
are  2  and  2?"  etc.  Of  course  for  a  time  some  of  these 
pupils  answered  "4"  to  the  2  and  2,  not  because  they  per- 
ceived the  identity  of  the  2  with  the  i  and  i,  but  because 
the  "4"  was  the  last  answer  that  they  had  heard.  It  was 
not  long,  however,  before  almost  unconsciously  the  per- 
ception came  and  they  were  able  to  apply  the  resulting 
principle  in  finding  other  sums. 

Very  persistent  repetition  was  necessary  in  the  case  of 
these  "slow  pupils,"  to  insure  the  thorough  memorizing  of 
the  facts  thus  derived.  Again  and  again  did  the  teacher 
judge  that  the  association  had  been  firmly  made,  only 
to  discover  when  she  somewhat  relaxed  the  drill  that  her 
faith  had  not  been  well  founded.  Had  the  counting 
proved  dull  and  tiresome  to  the  child,  opponents  of  num- 
ber teaching  during  the  first  year,  would  have  found  addi- 
tional argument  in  the  case  of  this  lowest  third  of  the 
beginners.  On  the  contrary,  the  counting  continued  to 
be  for  all  pupils,  from  first  to  last,  a  continual  pleasure. 
They  loved  the  rhythm,  they  gloried  in  conquering  one 
series  after  another,  as  they  glory  in  achievements  in  their 
play.  The  satisfaction  with  which  they  counted  by  5's  or 
6's,  and  the  contempt  but  half  assumed  with  which  they 
who  were  thus  advanced,  would  rush  through  I's  and  2's, 
settled  once  and  for  all  the  fear  that  the  interest  shown  in 
counting  by  ones  might  not  continue  in  the  higher  series. 

Although  after  a  careful  attempt  had  been  made  in 
every  school,  the  pupils,  including  the  great  majority  of 


73 

beginners,  had  experienced  little  difficulty  in  learning  to 
count  backwards — less  difficulty  of  course  than  they 
would  have  had  in  learning  an  entirely  new  series, — when 
it  came  to  drill  on  individual  facts,  the  number  who  per- 
sistently confused  the  sums  and  differences,  was,  as  had 
been  anticipated — too  considerable  to  be  ignored.  It  was 
plainly  unpsychological  to  have  a  child  who  found  some 
difficulty  in  so  simple  a  verbal  association  as  that  between 
a  given  number,  "and"  and  the  digit  next  succeeding, 
judge  between  such  association  and  that  formed  between 
the  same  number,  "less,"  and  the  digit  immediately  preced- 
ing it.  For  many  the  judgment  was  too  complex.  Some 
first  giving  the  sum  and  seeing  disapproval  in  the  teacher's 
eye,  would  give  the  difference.  Others  of  different  temp- 
erament, or  more  immature,  familiar  with  both  facts  but 
confused  by  the  sudden  demand  for  a  judgment,  would  re- 
main in  a  state  of  uncertainty,  undoubtedly  painful,  until 
relieved  by  kindly  suggestion  or  stunned  by  sharp  rebuke. 

While  a  fair  majority  of  the  beginners  and  a  very  con- 
siderable majority  of  the  other  pupils  made  the  neces- 
sary discrimination  without  apparent  difficulty,  as  has 
been  already  asserted  there  seemed  to  be  no  reason  why 
the  minority  should  find  a  serious  obstacle  to  progress 
in  an  operation  which  was  by  no  means  a  necessary  condi- 
tion to  the  more  rapid  progress  of  the  more  mature.  As 
it  still  appeared  highly  probable  that  the  former  might  be 
more  able  to  judge  between  the  two  sets  of  facts  later  on 
in  the  school  year,  after  many  of  the  sums  had  been  cer- 
tainly mastered,  the  teaching  of  the  elementary  differences 
was  abandoned  in  every  school  but  one. 

This  action  found  additional  justification  from  another 
consideration  already  brought  out  in  the  a  priori  discus- 
sion. Early  in  the  school  year,  every  class  had  been 
asked  and  had  readily  answered  the  following  question: 
"If  I  had  an  apple  and  a  peach,  and  took  away  the  peach, 
what  would  I  have  left?  If  I  had  the  apple  and  the  peach 
and  took  away  the  apple?  All  but  an  inconsiderable 


74 

minority  were  equally  successful  in  answering  the  follow- 
ing: "I  have  some  fruit,  an  apple  and  a  peach.  If  I  take 
the  apple  away  from  the  fruit,  what  will  I  have  left?" 
etc.  When  numbers  were  substituted  for  apples  and 
peaches  in  the  first  question,  the  answer  was  given  quite 
as  readily.  When,  however,  the  sum  of  the  two  numbers 
was  substituted  for  the  term  "fruit"  in  the  second  ques- 
tion, a  majority  of  the  answers  were  confused, — it  being 
of  course  quite  natural  for  the  "slower"  children  to  give 
the  sum — the  number  last  named — in  place  of  the  re- 
mainder. Nevertheless,  with  the  aid  of  concrete  illustra- 
tion and  tactful  questioning,  even  this  difficulty  was  over- 
come, by  so  large  a  percentage  of  the  pupils,  that  it  seemed 
quite  possible  that  with  proper  training,  the  great  ma- 
jority might  be  led  to  form  the  generalization,  that  when 
one  of  two  numbers  is  taken  from  their  sum,  the  remainder 
will  always  be  the  other.  As  this  generalization  is 
"effective,"  in  that  through  it  all  fundamental  differences 
can  be  readily  derived,  the  steps  necessary  to  it  were  from 
this  time  on  made  a  part  of  the  regular  drill.  The  gen- 
eral test  which  will  be  presently  described,  having  been 
taken  too  early  in  the  year  for  the  results  of  this  post- 
ponement and  preliminary  drill  to  be  statistically  reported, 
it  may  be  well  to  state  that  the  experiment  was  highly 
successful.  Half,  for  example,  of  the  pupils  in  one 
advanced  class  at  the  close  of  the  first  half  year  gave 
with  absolute  correctness  and  with  very  considerable 
readiness  any  fundamental  differences  corresponding  to 
fundamental  sums  which  they  had  mastered.  This  in- 
formal test  was  as  much  of  a  surprise  to  the  teacher  as  to 
the  pupils,  and  so  far  as  was  known,  involved  no  fact 
which  the  class  had  previously  derived.  A  majority  of 
the  pupils  still  failing,  however,  to  master  the  generaliza- 
tion, it  was  found  necessary  for  them  to  derive  the  differ- 
ences as  they  had  derived  the  sums, — the  backward  count- 
ing proving  as  popular  and  successful  a  process  as  count- 
ing forwards. 


75 

The  abstract  derivation  of  each  new  sum,  involving 
as  it  does,  the  oral  addition  of  three  numbers,  was  the 
beginning  of  a  thorough  preparation  for  rapid  operation. 
Wherever  the  teachers  strictly  excluded  all  examples 
involving  facts  insufficiently  mastered  and  persistently 
carried  on  the  various  forms  of  preliminary  drill,  the 
habit  of  rapid  operation  was  formed  by  the  pupils.  While 
there  is  at  present  no  way  by  which  this  can  be  satisfac- 
torily demonstrated  to  those  who  have  not  visited  the 
schools,  it  is  none  the  less  a  fact  that  the  pupils  thus  taught 
do  not  know  what  it  is  to  add  or  to  subtract  slowly.32 

III.  NUMBER  TEST  GIVEN  AT  THE  CLOSE  OF  THE  FIRST 
HALF  YEAR  TO  237  FIRST  GRADE  PUPILS. 

(I)   OBJECT  OF  THE  TEST. 

The  test  was  made  to  determine: 

1.  The  percentage  of  the  whole  number  of  pupils  as 
well  as  of  the  beginners,  who  had  thoroughly  mastered  the 
facts  of  each  series,  as  shown  in  the  percentage  of  correct 
answers  immediately  given. 

2.  The  nature  of  the  deficiency  existing  among  the  re- 
maining pupils,  as  shown  by  the  percentage : 

(1)  Of  correct  answers  deliberately  given. 

(2)  Of  wrong  answers  immediately  corrected. 

(3)  Of  correct  answers  immediately  given  and  un- 
corrected. 

(4)  Of  incorrect  answers  immediately  given  and  cor- 
rected wrongly. 

( 5 )  Of  incorrect  answers  deliberately  given. 

(6)  Of  failures  to  answer. 

3.  The  comparative  readiness  with  which  the  facts 
were  given,  as  shown  by  the  average  time  it  took  the 
pupils  of  each  class  to  give  a  fact  in  each  series, — deduc- 
tion being  made  for  all  deliberations. 

That  is,  the  tests  sought  to  discover  how  thoroughly 


76 

the  fundamental  sums  had  been  mastered  and  how  readily 
they  could  be  called  to  mind  and  expressed. 

Immediate  answers  deliberately  given,  indicated  a 
mastery,  mechanical  to  a  higher  or  lesser  degree,  as  shown 
in  the  average  time  it  took  to  give  a  fact  in  each  group. 
The  comparison  of  the  time  thus  recorded  with  the  records 
resulting  from  future  tests  given  under  the  same  general 
conditions,  will  afford  the  measure  of  the  gain  or  loss  in 
mechanical  facility.33 

Correct  answers  deliberately  given  indicate  the  power 
to  rationally  derive  an  unknown  or  forgotten  fact  from 
one  that  is  known.  Wrong  answers  immediately  cor- 
rected are  more  likely  the  result  of  nervous  strain  due  to 
the  test,  or  of  psychological  or  physiological  conditions 
peculiar  to  an  individual,  than  of  ignorance  of  the  fact 
required.  Incorrect  answers  immediately  given  and  un- 
corrected  while  not  necessarily  indicative  of  individual 
ignorance,  make  collectively  a  fair  index  of  general  inac- 
curacy. While  incorrect  answers  immediately  given  and 
corrected  wrongly,  may  be  due  to  nervousness,  as  is  prob- 
ably the  case  with  those  which  are  corrected  rightly,  they 
are  more  likely,  perhaps,  to  result  from  uncertain  knowl- 
edge. Incorrect  answers  deliberately  given,  probably 
for  the  most  part  result  from  some  error  in  the  process 
of  derivation  or  from  the  basing  of  that  process  upon  an 
incorrect  sum  or  difference,  although  they  may  originate 
in  an  effort  to  supply  by  a  guess  a  forgotten  fact  that  can 
not  be  rationally  derived.  Failures  to  answer,  though 
frequently  due  to  embarrassment,  are  here  assumed  to  be 
the  result  of  ignorance. 

(II)   DESCRIPTION  OF  TEST  AND  OF  PRECAUTIONS  TAKEN 
TO  INSURE  TRUSTWORTHY  RESULTS. 

Both  in  the  classification  just  given,  as  well  as  in  the 
method  of  conducting  the  test,  every  effort  was  made  to 
prevent  the  more  or  less  unconscious  modification  of  con- 


77 

ditions,  so  often  due  to  the  conducting  of  an  investigation 
by  those  interested  in  its  results.  Errors  due  to  partiality 
in  questioning, — that  is,  to  the  teacher's  consciously  or  un- 
consciously asking  the  easier  facts  of  a  series  from  the  less 
proficient  pupils  and  the  more  difficult  from  those  who  are 
brighter,  were  avoided  by  the  adoption  of  a  fixed  order  for 
the  facts  of  each  series.  This  order  was  purely  arbitrary, 
and  included  all  sums  in  each  series,  or  their  inversions. 
In  order  that  the  test  might  measure  the  actual  progress 
of  the  schools,  the  facts  in  each  series  were  given  together. 
Test  was  also  made  of  the  facts  miscellaneously  grouped. 

The  lack  of  uniformity  in  the  rate  of  questioning, 
destructive  to  the  value  of  the  time  tests,  was  reduced  to 
a  minimum  by  all  facts  being  demanded  by  the  same 
individual  in  as  quick  succession  as  distinctness  of  enunci- 
ation would  allow,  and  for  a  period  so  limited  as  to  prac- 
tically eliminate  variations  due  to  fatigue.  Whenever 
a  pupil  hesitated,  deduction  was  made  in  the  time — all 
answers  appreciably  longer  in  coming  than  the  others 
being  counted  hesitations.  In  order  to  insure  accuracy 
and  to  prevent  interference  with  the  recording  of  results, 
the  time  was  kept  by  an  assistant.  The  teacher  took  no 
part  in  the  test  except  to  note  the  names  of  the  pupils  that 
failed,  and  the  probable  explanation  of  their  failure, 
where  explanation  could  be  given.  The  use  of  an  exceed- 
ingly simple  system  of  recording  made  it  possible  for  the 
individual  who  did  the  questioning  to  note  each  failure  in 
the  proper  class  without  loss  of  time, — his  pencil  being 
ever  ready  to  mark  a  tally  under  "immediate  answers," 
while  "deliberations"  afforded  time  for  the  necessary 
record. 

All  children  present  in  the  schools  on  the  day  when  the 
test  was  made, — with  the  exception  of  one  who  had  just 
entered  school, — were  subjected  to  it — 237  in  all.  Of 
these  but  68  were  beginners  who  had  been  in- 
cluded in  the  original  100  tested,  although  many  of  the 
others  had  entered  school  since  the  beginning  of  the 


78 

school  year.  Each  child  was  tested  on  one  fact  in  every 
group  which  his  class  was  thought  to  have  mastered.  The 
total  number  of  facts  demanded  from  the  whole  number  of 
children  was  1,838. 


(Ill)  ANALYSIS  OF  THE  RESULTS. 

i.  More  Facts  Were  Mastered  by  the  Pupils  Than  in 
Any  Corresponding  Period  in  Previous  Years. 

Though  the  test  was  severe,  its  results  fully  justified 
the  order  and  methods  that  had  been  followed  in  the  work 
of  the  first  half  year. 

In  the  past  it  was  only  the  most  advanced  classes  that 
had  mastered  the  twenty-five  elementary  sums  and  the 
twenty-six  elementary  differences,  involved  in  the  study 
of  the  numbers  from  i  to  10, — fifty-one  facts  in  all,  to- 
gether with  their  inversions  and  alternations.  A  few 
products  and  quotients  and  the  differences  resulting  from 
the  subtraction  of  each  digit  from  itself  are  not  included  in 
this  comparison, — their  mastery  having  been  incidental  to 
both  plans  of  work. 

Under  the  new  system,  16  per  cent  of  all  the  pupils 
and  15  per  cent  of  the  beginners  had  mastered  112 
fundamental  sums  with  their  inversions — over  twice  as 
many  facts  as  any  class  had  previously  mastered  in  the 
same  time,  and  in  addition  could  count  by  lo's  to  100, 
could  instantaneously  add  any  number  from  i  to  10  to  any 
multiple  of  10  below  100,  could  add  i  to  any  number 
from  i  to  100,  and  in  many  cases  could  immediately  derive 
all  facts  in  subtraction  corresponding  to  the  sums  which 
they  had  mastered.  Twenty-seven  per  cent  of  all  pupils 
and  21  per  cent  of  the  beginners,  were  as  far  advanced  as 
those  just  reported,  except  that  they  had  mastered  but 
ninety-nine  fundamental  sums,  and  that  it  was  a  smaller 
percentage  that  could  readily  derive  the  differences.  Re- 


79 

spectively  39  per  cent  and  31  per  cent  had  mastered  85 
facts ;  52  per  cent  and  34  per  cent,  70  facts ;  56  per  cent  and 
41  per  cent,  54  facts ;  and  79  per  cent  and  66  per  cent  had 
mastered  37  facts.  Seventy-two  per  cent  of  all  pupils 
and  52  per  cent  of  the  beginners  could  add  i  to  any  num- 
ber from  i  to  i  oo;  76  per  cent  and  63  per  cent  respectively 
could  add  any  digit  to  any  multiple  of  10  below  100,  and 
68  per  cent  and  44  per  cent  could  similarly  add  10. 
Eighty-five  per  cent  and  78  per  cent  respectively  could 
count  intelligently  by  ID'S  to  100,  and  all  could  add  i  to 
any  number  from  i  to  20.  No  facts  have  been  included 
in  these  figures  upon  which  the  pupils  were  not  thor- 
oughly tested.  The  majority  of  the  classes,  at  the  time 
the  test  was  made,  had  been  more  or  less  thoroughly 
drilled  on  the  series  next  above  the  highest  upon  which 
they  were  tested,  while  several  classes  had  begun  the 
systematic  derivation  of  the  fundamental  differences. 

It  should  be  noted  that  the  contrast  between  the  knowl- 
edge possessed  by  the  most  advanced  classes  in  this  and 
former  years,  was  no  greater  than  that  existing  between 
the  attainments  of  the  slowest  groups.  In  the  past  it  was 
a  source  of  gratification  to  the  teacher  if  the  unfortunates 
of  which  these  groups  were  composed  had  mastered  more 
or  less  thoroughly  the  first  eight  or  ten  facts.  Under 
the  new  plan  no  group  had  mastered  less  than  nineteen 
sums,  while  a  large  proportion  of  "slow"ones  had  mas- 
tered the  multiples  of  10,  could  add  the  digits  to  them, 
and  could  add  i  to  any  number  below  100. 

Such  results,  attained  under  unsatisfactory  conditions, 
and  in  spite  of  the  blunders  ever  involved  in  a  new  at- 
tempt, would  seem  to  indicate  that  the  plan  thus  tested, 
properly  carried  out,  will  make  possible  either  the  mas- 
tery of  more  number-facts  during  the  first  school  years, 
or  what  is  perhaps  more  desirable, — the  mastery  of  the 
customary  number  of  facts  with  less  time  devoted  to  the 
study  of  number. 


8o 

2.  The  Test  Showed  a  Remarkably  High  Percentage 
of  Correct  Answers. 

Notwithstanding  the  fact  that  every  child  present  in 
the  various  schools, — with  the  exception  already  noted — 
were  subjected  to  the  test,  82.1  per  cent  of  the  1,838  facts 
called  for  were  given  correctly.  As  in  the  17.9  per  cent 
of  wrong  answers  are  included  those  of  feeble-minded, 
backward,  or  nervous  children,  and  of  pupils  who  had 
been  but  a  few  weeks  in  school — some  of  whom  failed  on 
most  of  the  questions  and  a  few  on  all — the  true  per- 
centage of  correct  answers  should  be  much  higher.  For 
example,  in  one  school  of  forty  pupils,  four  children  made 
thirty-four  out  of  the  eighty  mistakes.  With  these  four 
eliminated,  the  percentage  of  correct  answers  for  the 
school  increases  from  80.5  per  cent  to  87.2  per  cent.  In 
the  same  ratio  the  82.1  per  cent  of  correct  answers  for  all 
schools  would  be  increased  to  89  per  cent. 

3.  The  Pupils  Displayed  a  High  Degree  of  Certainty 
in  Their  Answers. 

Only  1.3  per  cent  of  the  answers  were  changed,  when 
once  they  had  been  given — i  per  cent  of  these  being 
wrong  ones  that  were  corrected,  and  .3  per  cent  correct 
ones  that  were  made  wrong.  In  most  cases  these  errors 
were  the  result  of  nervousness. 

4.  The  Answers  Were  Given  with  Highly  Satisfactory 
Promptness. 

As  the  average  time  taken  in  asking  the  questions  dis- 
tinctly is  included  in  the  average  number  of  seconds  given 
in  the  following  table,  it  has  been  thought  best  to  give 
for  comparison  the  corresponding  averages  resulting, 
when  the  individual  who  asked  the  questions  in  the  test, 
asked  them  aloud  to  himself  and  himself  gave  the  an- 
swers with  maximum  readiness.  In  each  case  this  average, 


8i 

which  for  convenience  will  be  called  the  "standard,"  is 
the  lowest  resulting  from  three  of  these  self-tests.  The 
results  are  given  for  each  of  the  three  or  four  sub-groups 
into  which  the  number  of  classes  was  divided. 

Stand.  IstGr.  2d  Gr.  3d  Gr.  4th  Gr. 

I.  5  &  1,  1  &  2,  etc 2.25  2.26  2.62  2.99  2.59 

II.  6  tens,  2  tens,  etc 2.20  2.15  2.18  2.18  2.59 

III.  60  &  10,  10  &  10,  etc.        2.55  2.46  2.43  2.48  2.77 

IV.  20  &  4,  40  &  3,  etc.     ...   2.48  2.48  2.60  3.13  5.50 
V.  26  &  1,  52  &  1,  etc.  . . .     2.60  2.58  2.75  2.89 

VI.  6  &  2,  2  &  2,  etc 2.50  2.50  2.85  2.96 

VII.  12  &  3,  3  &  3,  etc 2.55  3.33  3.43  3.61 

VIII.  Miscellaneous  Sums     . .  2.40  3.13  3.39  2.50 

IX.  Miscellaneous  Inversions  2.45  3.38  3.94  3.75 

X.  12  &  4,  4  &  4,  etc 2.50  3.48  4.04  3.25 

XI.  10  &  5,  5  &  5,  etc 2.50  3.02  3.56  4.37 

XII.  12  &  6,  6  &  6,  etc 2.50  4.28  4.38  4.12 

XIII.  14  &  7,  7  &  7,  etc 2.50  3.91  3.57 

It  will  be  noted  that  in  most  cases  the  variation  is  not 
great, — much  of  it,  notwithstanding  all  precautions, — 
being  in  all  probability  due  to  failure  to  question  at  a  uni- 
form rate,  although  the  number  of  seconds  for  the  first 
group  is  uniformly  lower  than  those  of  the  second,  and 
those  of  the  second  than  those  of  the  third.  For  the 
first  six  series,  the  answers  are  given  with  a  high  degree 
of  readiness.  From  the  seventh  series  on,  there  is  a  very 
marked  decrease  in  the  readiness  with  which  the  answers 
are  given,  due  to  the  fact  that  the  pupils  in  most  of  the 
classes  had  been  given  less  drill  upon  the  higher  sums. 

The  averages  for  series  VIII,  in  which  the  "2"  and  the 
"3  sums"  are  mingled,  are  higher  than  those  of  series  VI, 
but  lower  than  those  of  series  VII, — the  indication  being 
that  the  pupils  could  give  the  miscellaneous  facts  as  readily 
as  those  of  the  groups.  The  fact  that  the  average  number 
of  seconds  in  series  IX, — where  the  facts  of  series  VIII 
are  inverted,  is  higher  than  that  in  series  VIII,  is  probably 
due  not  to  any  less  readiness  in  giving  the  ordinary  inver- 
sions, but  to  the  unfortunate  fact  that  it  contained  such 


82 

inversions  as  3  and  12,  2  and  16,  3  and  13,  and  2  and  14, 
which  are  rarely  called  for  either  in  addition  or  in  drill. 

On  the  whole,  the  facts  were  given  with  a  highly  satis- 
factory degree  of  readiness,  especially  when  it  is  taken 
into  account  that  the  main  reason  for  thus  timing  the 
pupils  at  a  stage  of  their  work  when  but  comparative 
little  drill  had  been  given  them  in  operation,  was  to  make 
it  possible  to  gauge  the  improvement  that  should  result 
by  the  close  of  the  year.  It  is  altogether  likely,  however, 
that  the  variation  due  to  oral  questioning  will  in  the  end 
prevent  a  satisfactory  comparison.83 

5.  Demonstration  of  the  Ability  of  the  Pupils  to 
Promptly  Derive  for  Themselves  Facts  Upon  Which 
They  Had  Been  Insufficiently  Drilled. 

This  evidence  of  independence  in  derivation  was  one 
of  the  most  significant  of  the  results.  When  pupils 
hesitated,  3.2  per  cent  of  the  questions  received  no  reply 
at  all — owing  to  nervousness,  dullness,  or  mental  con- 
fusion; 4  per  cent  after  deliberation  were  answered 
wrongly,  either  through  their  derivation  being  based 
upon  an  error,  or  through  some  mistake  in  the  process  of 
derivation,  unless  the  children  thus  answering  waited  a 
moment  or  so  to  give  a  guess  that  might  just  as  well  have 
been  given  at  once.  Nine  and  four-tenths  per  cent,  how- 
ever, were  answered  correctly  after  more  or  less  delibera- 
tion. If  the  facts  had  been  taught  in  Grubean  isolation 
such  recall  would  have  been  impossible.  The  significance 
of  the  foregoing  percentage  is  largely  due  to  the  fact  that 
the  pupils  who  thus  hesitated,  were  for  the  most  part 
those,  who  being  behind  the  majority  of  their  fellows, 
might  be  supposed  to  be  less  likely  to  abstractly  derive 
the  facts  that  they  had  forgotten. 

It  should  be  remarked  in  conclusion  that  throughout 
the  test,  the  beginners  showed  as  great  accuracy,  readi- 
ness and  independence  in  giving  such  facts  as  they  had 


83 

mastered,  as  did  the  pupils  who  had  attended  school 
before. 

IV.  SUMMARY    OF    THE    CONCLUSIONS    WHICH    THE 
INQUIRY  TENDS  TO  ESTABLISH. 

At  the  close,  then,  of  the  first  half  year  of  work,  the 
results  attained  all  tend  to  demonstrate  the  correctness  of 
the  a  priori  determinations  regarding  psychological  order 
and  method.  While  both  are  psychological  only  in  so  far 
as  they  are  adapted  to  the  majority  of  individuals  taught; 
while  it  is  altogether  likely  that  there  are  individuals  to 
which  they  are  not  thus  adapted ;  they  have  proved  them- 
selves psychological  in  the  only  schools  where  thus  far 
they  have  been  tried.  Nor  is  it  unreasonable  to  assume 
that  based  as  they  are  upon  psychological  fact,  should  they 
be  adopted  by  other  schools,  they  may  in  course  of  time 
be  as  certainly  confirmed  by  general  experience  and 
proven  by  scientific  test,  as  they  have  been  indicated  by 
individual  reason  and  justified  by  individual  success.  For 
the  present,  a  single  investigator  tentatively  asserts  as  the 
result  of  a  patient  and  impartial  examination  of  limited 
data,  that  the  only  numerical  knowledge  common  to  a 
majority  of  children  on  entering  school  is  the  ability  to 
instantaneously  perceive  three  or  four  objects  as  3  or  4, 
and  to  count  more  or  less  mechanically  to  10  or  12;  that 
the  majority  of  such  children  love  to  count,  and  as  their 
work  progresses — to  count  by  2's,  3's,  etc.,  as  well  as  by 
I's;  that  this  knowledge  and  this  interest  form  the  natural 
basis  for  the  study  of  number ;  that  there  is  no  necessary 
antagonism  between  the  logical  and  the  psychological 
orders  of  teaching  the  fundamental  sums  and  differences ; 
that  the  order  in  which  they  can  be  most  readily  taught  is 
that  logical  order  which  results  from  the  successive  addi- 
tion or  subtraction  of  each  digit  from  all  other  digits, — 
in  that  each  of  the  resulting  groups  contains  an  element 
which  readily  mastered  in  one  or  two  of  the  facts  to  which 


84 

it  is  common,  readily  effects  the  derivation  of  all ;  that  this 
method  of  derivation  involves  the  maximum  of  abstract 
reason  possible  to  the  majority  of  children  on  first  enter- 
ing school  and  the  minimum  of  concrete  illustration  neces- 
sary to  intelligent  work ;  that  it  is  based  upon  the  numeri- 
cal knowledge  possessed  by  beginners,  insures  from  the 
start  a  maximum  amount  of  the  mental  training  peculiar 
to  mathematics,  and  makes  possible  the  use  of  abstract 
counting  by  2's,  3*5,  etc., — the  readiest  means  of  memoriz- 
ing the  sums  and  differences  when  once  they  are  derived ; 
that  the  order  and  the  method  which  thus  result  in  the 
readiest  mastery  and  the  highest  training  are  mathemati- 
cal, and,  being  adapted  to  the  great  majority  of  the  pupils 
of  even  the  first  school  grade,  psychological;  that  since 
their  proper  use  insures  the  mastery  of  more  funda- 
mental sums  and  differences  in  the  usual  time  or  of  the 
usual  number  of  sums  and  differences  in  less  time,  and 
gives  the  pupils  the  power  to  readily  derive  forgotten 
facts,  they  are  economical ;  that  rapid  operation  in  greater 
or  less  degree,  can  be  made  invariable  from  the  first,  and 
so  from  the  first  can  become  a  habit. 


NOTES  AND  REFERENCES. 

1  "  Vorstellungkreis  der  Berliner  Kinder  beim  Eintritt  in  die  Schule." 
Berlin  Stadisches  Jahrbuch.     1870.     Pp.  59-77. 

2  "Contents  of  Children's  Minds  on  Entering  School,"  by  G.  Stanley 
Hall,  Ph.  D.,  LL.  D. 

3  Ibid.,  p.  12.     Dr.  Hall  here  refers  to  "  Der  Vorstellungskreis  unserer 
sechsjahrigen    Kleinen."     Allg.  Schul-Zeitung.     Jena,    1879.    P.    327 
et  seq.     Tliis  article  I  have  been  unable  to  consult. 

4  "Contents  of  Children's  Minds,"  p.  19. 

5  Ibid.,  p.  34. 
9  Ibid.,  p.  17. 

7  Ibid.,  p.  49. 

8  This  conclusion,  while  merely  tentative  in  so  far  as  it  is  based  upon 
the  foregoing  results,  is  in  accord  both  with  common  experience  and 
the  general  trend  of  philosophical  determination.     The  Committee  of 
Fifteen  asserts  that  "counting  is  the  fundamental  operation  of  arith- 
metic, and  all  other  arithmetical  operations  simply  devices  for  speed 
by  using  remembered  countings  instead  of  going  through  the  detailed 
work  again  each  time."     ("Report  of  the  Committee  of  Fifteen,"  pub- 
lished for  the  National  Educational  Association  by  the  American  Book 
Company.    New  York,  Cincinnati,  Chicago,  1895,  p.  53.) 

Believers  in  the  Culture  Epoch  Theory  can  not  but  accept  this  asser- 
tion, on  the  ground  that  in  the  historical  development  of  the  race, 
counting  preceded  all  other  forms  of  numerical  operation.  ("The  Teach- 
ing of  Elementary  Mathematics,"  D.  E.  Smith.  Macmillan,1900,  p.  49.) 
It  finds  its  fullest  justification,  however,  in  Dr.  D.  E.  Phillips'  article 
on  "Number  and  Its  Application,  Psychologically  Considered,"  in 
"Pedagogical  Seminary,"  Vol.  V,  No.  2,  pp.  221-281.  (J.  H.  Orpha, 
Worcester,  Mass.)  "The  earliest  and  most  rudimentary  form  of 
knowledge  in  the  cognitive  sense,"  says  Dr.  Phillips,  "is  a  knowledge 
of  a  series  of  changes"  (p.  228).  This  "series-idea,"  established  by 
a  multitude  of  successive  and  rhythmical  sensations  conveyed  through 
the  different  senses,  and  embodying  the  rudiments  of  the  number 
concept,  finally  becomes  abstract — a  general  idea  applicable  to  any 
series  of  successions.  At  this  stage  symbols  of  representation  are  neces- 
sary. Hence  results  counting,  first  the  naming  of  the  series  without 
reference  to  objects  of  any  kind,  then  the  counting  of  objects  inde- 
pendently of  the  order  of  number-names,  and  finally  the  counting  of 
objects  with  the  number-names  applied  in  proper  order. 

"Counting  is  fundamental,  and  counting  that  is  spontaneous,  free 
from  sensible  observation,  and  from  the  strain  of  reason"  (p.  238). 

(85) 


86 

9  Such  records  as  investigators  have  made  of  the  mental  develop- 
ment of  individual  children  fully  confirm  this  statement.     Perez  reports 
the  case  of  a  child  two  and  a  half  years  old  who  could  count  to  12,  but 
did  not  understand   the   expression   "3   days"   until   "another  and 
another  and  another"  made  it  clear.     To  the  same  child  a  year  was- 
"many,  many  to-morrows."     ("The  First  Three  Years  of  Childhood," 
Bernard  Perez.     Edited  and  translated  by  Alice  M.  Christie.     E.  L. 
Kellogg  &  Co.,  1894,  p.  186.) 

Similarly  in  a  case  reported  by  Preyer,  by  the  twenty-ninth  month 
numbering  begins  to  be  active  to  a  noteworthy  degree.  "The  child 
began,  viz,  on  the  878th  day,  suddenly  of  his  own  accord  entirely,  ta 
count  with  his  nine-pins,  putting  them  in  a  row,  saying  with  each  one, 
eins  (one)  eins!  eins!  eins!  eins!  Afterward  saying,  eins!  noch  eins 
(one  more) !  noch  eins !  noch  eins !  The  process  of  adding  is  thus 
performed  without  the  naming  of  the  sums."  ("The  Development 
of  the  Intellect,"  W.  Preyer.  Appleton,  New  York,  1895,  p.  172.) 

10  Among  453  children  first  entering  the  schools  of  Chester,  Pa.,  in 
September,  1900,  but  56  (i.  e.,  12 £  per  cent)  were  found  unable  to 
count  to  8  or  10. 

11  While  Preyer's  generalization  that  "  the  boy  of  four  years  counts 
objects  with  effort  up  to  six"  and  that  "numbers  remain  for  a  long 
time  merely  empty  words"  (Dev.  of  Int.,"  p.  271),  may  be  based  upon- 
too  few  particulars,  the  latter  statement  at  least  is  undoubtedly  correct. 
(See  Dr.  Phillips'  "No.  and  Its  Ap.,"  p.  234.)     Cases  where  "counting 
proceeds  independently  of  the   number-names"  (Phillips,  p.  234),  as 
where  a  child  of  nineteen  months  counts  "2,  3,  5,  6,  7,  8,  9"  (Preyer), 
while  probably  characteristic  of  "at  least  nearly  all  children"  when 
they  first  learn  to  count,  are  by  no  means  common  among  the  children 
whom  I  have  tested  on  their  first  entering  school — probably  less  than 
half  a  dozen  in  several  hundred.     The  inference  is  plain;  most  children 
in  all  probability  begin  to  count  long  before  they  reach  school  age. 

12  The  fact  that  in  counting  objects,  children  often  run  the  number 
series  ahead,   counting  "seven"   before  the  teacher  has  placed  the 
seventh  object  with  the  five  or  six,  should  not  be  taken  to  indicate 
inability  to  number  objects.     The  error  is  due  rather  to  a  lapse  in 
attention,  and  is  not  so  likely  to  occur  when  the  pupils  are  themselves 
handling  the  objects  which  they  count. 

13  By  "intelligent"  counting,  I  mean  nothing  more  than  the  num- 
bering of  things  successively  perceived,  by  a  parallel  repetition  of  the 
accepted  number  series,   accompanied  by  the  realization  that  each 
number-name  applies  to  one  thing  more  than  that  which  immediately 
precedes  it  in  the  scale. 

Of  course  this  realization  of  the  inclusiveness  of  each  number-name 
involves  the  discrimination,  abstraction  and  grouping  justly  empha- 
sized by  Professor  Dewey  (McLennan  and  Dewey's  "Psychology  of 


87 

Number,"  Appleton,  1898,  pp.  31  and  32),  the  number-name  denoting 
not  the  one  more,  but  the  number  of  ones  in  the  group  of  which  the 
one  more  forms  a  part.  But  when  children  who  have  mastered  the 
number  series  abstractly,  apply  it  to  objects  in  order  to  determine  "  how 
many"  it  is  under  conditions  which  insure  the  presence  of  each  factor 
in  the  intellectual  process. 

Professor  Dewey  asserts  that  in  order  "  to  promote  the  natural  action 
of  the  mind  in  constructing  number,  the  starting  point  should  be  not 
a  single  thing  or  an  unmeasured  whole,  but  a  group  of  things  or  a 
measured  whole."  So  far  as  the  "single  thing"  is  concerned,  he  might 
have  gone  a  step  farther.  The  starting  point  in  counting  not  only 
should  not  be  a  single  thing,  but  can  not  be  so.  Before  an  individual 
attempts  to  determine  "how  many,"  there  must  be  in  his  mind  the 
thought  of  the  unified  group  that  he  is  to  number.  Its  unity  need  not 
be  that  of  qualitative  likeness.  It  may  consist  in  the  mere  fact  of 
possession  or  location,  as  when  he  numbers  the  Christmas  presents 
that  he  has  received,  or  the  various  objects  on  a  table.  Ignoring  quali- 
tative differences  with  ease,  or  failing  to  observe  them,  abstraction 
presents  less  difficulties  to  him  than  to  the  philosopher.  He  unhesi- 
tatingly counts  things  that  the  philosopher  would  tend  to  subgroup. 
But  the  thought  of  some  constant  unity  such  as  the  number  of  legs 
possessed  by  a  horse,  or  of  a  constantly  increasing  unity  such  as  the 
telegraph  poles  seen  from  the  car  window,  always  precedes  his  count- 
ing. There  is  never  a  " how  many"  without  a  supplementary  "  what?" 
the  thought  of  which  precedes  it,  and  the  "what"  is  never  a  single 
thing,  but  always  a  unified  plurality. 

Just  as  generalization,  abstraction  and  grouping  thus  precede  the 
counting  of  things,  discrimination  acccompanies  it.  As  each  thing, 
observed  in  its  proper  sequence,  serves  as  a  stimulus  to  the  correspond- 
ing number-name,  it  must  be  clearly  perceived  as  a  one  of  right  belong- 
ing to  the  group.  If  things  can  not  be  discriminated  they  can  not  be 
counted.  It  therefore  follows  that  if  children  succeed  in  applying  the 
number  series  to  things  at  all,  they  are  grouping,  abstracting  and  dis- 
criminating like  so  many  philosophers.  Professor  Dewey  does  not 
give  too  philosophical  a  meaning  to  children's  counting  (Phillips,  p.  243) ; 
he  merely  describes  in  philosophical  terms  an  experience  which  children 
necessarily  have  whenever  they  apply  the  number  series  to  things,  but 
of  which  they  are — and  may  be  permitted  to  remain — unconscious. 
Before  they  attempt  to  determine  how  many,  they  generalize;  if  they 
count  things  at  all  they  discriminate  them. 

If  he  would  go  on  to  assert  that  the  naming  of  each  successive  number 
as  one  thing  more  is  added  to  the  group  already  counted,  carries  with 
it  a  recollection  and  discrimination  of  the  individual  things  already 
numbered  and  their  regrouping  with  the  additional  thing  into  the  new 
unity  to  which  the  number-name  is  applied,  I  should  venture  to  dispute 


88 

it  on  what  appears  to  me  fully  adequate  grounds.  (See  p.  61  of  present 
discussion.) 

That  there  is  a  vague  recollection  of  the  mass  of  number-names  that 
have  gone  before,  made  possible  in  the  case  of  the  larger  numbers,  by 
the  decimal  terminology,  I  believe  to  be  true.  When  in  counting  we 
say  "twenty-six,"  there  is  no  clear  recollection,  visual  or  otherwise, 
of  26  things  (see  p.  62  of  present  discussion),  but  rather  the  abstract 
thought  of  26  as  25  and  1,  and  the  vague  knowledge — perhaps  not 
present  in  every  instance,  but  always  possible — that  many  successive 
additions  of  1  were  necessary  before  the  first  1  was  increased  to  25. 
When  the  decimal  system  has  been  mastered,  added  to  this,  if  not 
entirely  displacing  it,  there  may  be  the  abstract  recall  of  two  tens  and 
six,  a  concept  fully  adequate  to  the  demands  of  every-day  life,  if  not 
to  mathematical  philosophy. 

A  vague  content  may  be  given  the  number-name  long  before  mechani- 
cal counting  becomes  intelligent.  Indeed,  Professor  LeFevre  asserts 
that  from  the  first  there  is  a  fuller  recall  of  number-names,  and  hence 
a  more  exact  notion  of  number  than  I  can  venture  to  claim.  On  p.  23 
of  his  "Number  and  Its  Algebra"  (Heath,  1896),  he  states  that  "the 
order  being  learned  by  rote,  any  word  numeral  by  suggesting  its  definite 
place  in  the  fixed  series  of  words,  recalls  all  those  gone  before;  and 
from  this  comparison  the  mind  conveys  or  receives  an  exact  notion  of 
the  number  of  individuals  in  the  group  of  objects  numerically  charac- 
terized by  any  such  number-name." 

14  P.  249,  "  Number  and  Its  Application." 

15  See  pp.  171-176  in  "  Outlines  of  Psychology,"  translated  by  Dr.  E. 
B.  Titchener  from   the  German  of   Professor  Oswald  Kiilpe.      (The 
Macmillan  Company,  New  York,  1895.) 

18  See  early  files  of  the  "Pennsylvania  School  Journal"  and  other 
educational  periodicals. 

17  See  "Elements  of  the  Grube Method,"  Levi  Seeley,  M.  A.,  Ph.  D. 
(E.  L.  Kellogg  &  Co.,  New  York.) 

18  Occasionally  the  general  order  is  varied,  as  where  2,  4  and  8  are 
analyzed  before  3,  5  and  7,  but  it  may  be  safely  asserted  that  in  almost 
all  American  schools  the  various  combinations  and  separatione  of  a 
particular  number  are  exhausted  before  the  teaching  of  any  fact  involv- 
ing the  numbers  above  it.     A  useful  piece  of  work  yet  to  be  accom- 
plished is  the  certain  determination  of  the  number  of  schools  using 
each  of  the  possible  orders  of  presenting  the  fundamental  number-facts 
and  of  the  various  general  methods  of  teaching  them. 

19  The  naturalness  of  such  derivation  as  well  as  its  rationality  is  seen 
in  the  following  abstract  from  Louise  E.  Hogan's  "Study  of  a  Child." 
(Harpers,  1898,  New  York,  p.  168.)     "To-day  we  heard  him  say  'six 
and  three  are  nine;   six  and  four  are  ten.'     His  aunt  asked  him  how 
he  knew  it.     He  replied,  'I  know  that  six  and  three  are  nine,  and  four 


89 

is  one  more  than  three,  and  ten  is  next  to  nine,  so  it  must  be  so.' "     (See 
Note  26.) 

30  A  summary  of  methods  given  in  David  Eugene  Smith's  admirable 
book,  "The  Teaching  of  Elementary  Mathematics"  (Macmillan,  1900, 
pp.  94-97),  leads  me  to  infer  that  this  plan  has  been  long  in  use  in  some 
of  the  elementary  schools  of  Germany.     He  refers  to  Knilling's  "Zur 
Reform  des  Rechenunterrichtes "  and  "Die  Naturgemasse  Methode  des 
Rechen-Unterrichts  in  der  Deutschen  Volksschule,"  and  to  Tanck's 
"Rechnen  auf  der  Unterstufe,"  "Der  Zahlenkreis  von  1  bis  20,"  and 
"  Betrachtungen  uber  das  Zahlen."     To  these  works  I  unfortunately 
have  not  had  access. 

31  Some  who  read  this  statement  may  prefer  to  agree  with  Dr.  Dewey, 
when,  after  asserting  that  "  the  concept  two  involves  the  act  of  putting 
together  and  holding  together  the  two  discriminated  ones,"  he  goes 
on  to  say,  "It  is  this  tension  between  opposites  which  is  largely  the 
basis  of  the  childish  delight  in  counting."     ("Psychology  of  Number," 
p.  31.)     It  seems  to  me,  however,  that  the  delight  is  in  mechanical 
counting — which  children  love  without  any  thought  of  the  fact  that 
one  and  one  are  two. 

While  Dr.  Phillips  finds  that  the  "passion  for  counting"  objects, 
most  common  in  children  between  the  ages  of  seven  and  ten,  had  its 
beginning  in  a  "passion  for  following  the  abstract  series"  in  but  57 
cases  out  of  131  (Ped.  Sem.,  V,  p.  252),  I  have  failed  to  discover  any 
children  of  school  age  who  do  not  heartily  enjoy  the  rhythmical  flow 
of  mechanical  counting  even  when  "sing-song"  is  prohibited. 

33  See  Phillips,  p.  265.     "Counting  is  fundamental  and  even  com- 
binations furnish  the  first  step,  hence  counting  by  2's,  3's,  4's,  etc., 
furnish  the  first  steps  in  multiplication." 

38  Encouraged  by  the  results  of  this  first  attempt  at  demonstration, 
during  the  past  school  year  (1900-1901),  I  followed  the  same  plan  in 
teaching  the  elementary  sums  and  differences  to  1,200  children  con- 
stituting the  first  grade  in  the  public  schools  of  Chester,  Pa.  While  I 
shall  from  time  to  time  refer  to  the  results  there  attained,  it  is  perhaps 
well  at  this  point  to  make  the  general  statement  that  this  second 
attempt  was  even  more  successful  than  was  the  first. 

34  See  Phillips,  p.  265 :   "  The  difficulties  of  subtraction  are  greater 
and  more  lasting  than  those  of  addition,  hence  its  introduction  on 
beginning  addition  must  at  first  be  incidental.     Care  must  be  taken 
not  to  confuse  the  child  by  introducing  too  many  processes  at  once." 
While  the  results  of  my  own  work  show  the  wisdom  of  Dr.  Phillips' 
caution  (see  present  article,  p.  48),  I  find  that  after  pupils  have  thor- 
oughly mastered  the  fundamental  sums  formed  by  the  addition  of  two, 
they  can  take  up  the  corresponding  differences  without  any  serious 
confusion  of  the  two  sets  of  facts. 

38  An  early  protest  against  the  "Bitter  End  of  the  Object  Method" 


9o 

was  made  by  Ida  F.  Foster  in  Educational  Review,  March,  1894.  "  It 
is  well  that  a  child  should  see  that  four  straws  and  three  straws  are 
seven  straws,  before  he  learns  to  say  that  4  +  3  =  7,  but  it  is  not  neces- 
sary that  four  splints  and  three  splints  should  be  set  before  his  eye 
every  time  he  is  asked  how  many  4  +  3  make."  Surely  someone, 
though  unknown  to  me,  must  have  long  ago  pointed  out  the  equal 
absurdity  of  objectively  deriving  each  new  fact. 

38  Hensleigh  Wedgwood,  in  his  article  on  "  The  Foundation  of  Arith- 
metic" ("Mind,"  Vol.  Ill,  p.  572),  declares  that  the  Grube  Method 
assumes  with  Mill  that  the  number-facts  are  mastered  only  "by  con- 
stant observation  of  the  result  when  groups  of  actual  objects  are  com- 
bined or  separated,  i.  e.,  by  experience."  Not  so,  he  goes  on  to  state, 
with  the  derivation  of  numbers  by  mental  association — "from  conscious- 
ness on  the  contemplation  of  our  own  thoughts,  that  what  we  mean 
by  two  is  nothing  else  than  the  aggregation  of  1  +  1."  When  with 
the  aid  of  objects,  children  have  learned  that  2  +  1  are  3,  and  that 
3  +  1  are  4,  they  can  be  quite  readily  led  without  other  aid  than  their 
own  powers  of  generalization  unconsciously  exercised,  to  the  conscious- 
ness of  the  fact  that  8  +  1  are  9.  This  statement  I  make  not  as  a 
reasonable  assumption,  but  as  an  assertion  based  upon  the  experience 
of  1,500  children. 

37  A  prolonged  effort  which  I  made  a  year  or  so  ago  to  train  twenty 
young  men  and  women  to  instantaneous  recognition  of  the  number 
8,    with   counting  and    estimation    both    conscious    and    unconscious 
practically  eliminated,   through  the  use  of  slightly   curved  lines  of 
equidistant  and  uniform  black  dots,  resulted  in  their  ready  recognition 
of   the  particular  7's,  8's  and  9's  on  which   they  had  been  drilled, 
but  in  no  appreciable  improvement  when  they  were  called  upon  to  dis- 
tinguish from  each  other,  8's,  7's  and  9's  arranged  in  curves  slightly 
different  from   those   used   in  their  training.      Since   with  irregular 
arrangement  I  found  training  in  recognition  quite  easy,  I  infer  that 
results  such  as  those  which  have  been  achieved  by  Miss  Aiken  (see 
"  Mind  Training" — Catherine  Aiken)  are  due  to  training  in  instantaneous 
and   perhaps   unconscious   estimation   rather   than   to    simultaneous 
number  perception. 

38  Children  often  distinctly  visualize  the  smaller  numbers  when  count- 
ing.    Of  this  Sully  gives  us  a  good  example — his  child  saying,  "One, 
two,  three,  four — two  dollies,  a  tin  soldier  and  a  shell,"  as  he  succes- 
sively pointed  to  their  images  on  his  coverlid.     (P.  484,  "  Studies  of 
Children,"  Appleton,  New  York,  1896.) 

To  what  limit  we  can  appreciate  the  number  of  units  involved  in  any 
written  or  spoken  quantity  is  problematical.  Dr.  Conant  thinks  100 
somewhat  high  for  most  of  us,  and  1,000  a  rather  certain  maximum. 
(P.  33,  "  The  Number  Concept,"  Macmillan,  New  York,  1896.)  Accord- 
ing to  Kay,  in  his  "Memory,  What  It  Is  and  How  to  Improve  It," 


91 

(p.  320,  Appleton,  New  York,  1895),  "G.  P.  Bidder,  before  he  was  six 
years  old,  was  accustomed  to  amuse  himself  by  arranging  peas,  marbles 
and  the  like  in  rows  and  squares  of  different  numbers,  and  by  counting 
them  over  to  ascertain  the  results  of  the  combinations.  Hence  the 
figures  did  not  present  themselves  to  him  merely  as  symbols,  but  they 
represented  to  his  mind  an  equal  number  of  definite  objects.  He  never 
went  beyond  100."  As  a  result,  however,  "he  could  multiply  a  row 
of  fifteen  figures  by  another  row  of  the  same  number  and  give  the  actual 
result  in  a  few  minutes  without  seeing  or  writing  down  a  single  figure." 
Such  use  of  imaged  objects  by  a  mathematical  prodigy — probably  as 
a  mnemonic  system — furnishes  no  justification  for  the  excessive  use 
of  splints  and  cubes  in  teaching  the  fundamental  number-facts  to  normal 
children. 

In  the  Educational  Review  for  1893  (p.  467),  Adelia  R.  Hornbrook 
claims  pedagogical  value  for  "number  forms,"  and  suggests  a  system 
of  instruction  based  upon  imaged  columns  of  the  numbers  written  in 
convenient  forms — a  device  which  in  many  cases  might  be  practi- 
cable, but  which  in  most  cases  would  be  unnecessary,  and  hence  uneco- 
nomical. 

28  Kay  asserts  ("Memory,"  p.  321)  that  if  the  fundamental  sums  are 
so  fully  associated  together  in  the  mind,  "that  afterwards,  whenever 
the  two  forms  occur,  the  third  being  their  sum,  will  at  once  come  up — 
one  may  sum  up  a  whole  column  of  figures  almost  at  sight.  This  is 
best  done  by  simply  using  the  eye  without  naming  the  figures."  That 
is,  the  column  3,  2,  1,  4,  etc.,  should  be  read  "  5,  6, 10, "  without  naming 
the  3  +  2,  the  5  +  1,  etc. 

But,  Kay  goes  on  to  say,  he  will  never  thoroughly  learn  the  combina- 
tion 9  +  8  =  17,  "if  he  immediately  adds,  and  6  =  23,  and  7  =  30. 
The  mind  must  dwell  upon  one  set  of  figures,  and  thoroughly  master 
it  before  proceeding  to  the  next." 

When  the  number-facts,  as  here  suggested,  are  taught  in  groups 
formed  by  the  successive  addition  of  the  same  digit — as,  for  example, 
"2  and  2,  4;  4  and  2,  6;  6  and  2,  8,"  etc. — it  is  the  introduction  by  the 
teacher  of  the  "and  2"  into  the  series,  which  thus  causes  the  mind  to 
dwell  upon  the  individual  facts,  and  so  completes  the  necessary  associa- 
tion. This  introduction  of  the  common  element  should  therefore 
invariably  supplement  the  simple  counting,  in  order  that  six  will  come 
after  four  in  the  children's  minds  only  when  they  are  counting  by  two's, 
and  after  five  when  they  are  counting  by  one's.  Then,  as  the  result  of 
the  mechanical  drill,  4  and  2  will  always  be  6,  and  not  5  or  7. 

30  Of  the  453  beginners  in  Chester,  12  per  cent  after  learning  to  count 
had  difficulty  in  telling  what  number  came  next  after  each  digit,  and 
15  per  cent  were  slow  to  learn  the  " one  sums"  to  ten. 

31  This  inference  has  been  borne  out  by  my  later  experience. 

33  In  order  to  achieve  this  end,  it  was  found  necessary  to  specially 


92 

prepare  for  the  teachers  the  examples  needed  for  proper  drill  and  to 
insist  upon  their  exclusive  use  in  seat  work  as  well  as  in  class  exercises. 
The  semi-oral  drill  upon  the  two  or  three  facts  other  than  the  "one 
sums"  involved  in  each  set  of  examples  was  invariably  insisted  upon 
before  the  pupils  were  permitted  to  add,  and  the  corresponding  written 
drill  before  they  were  allowed  to  subtract. 

After  examples  were  copied  for  desk-work,  the  pupils  added  or  sub- 
tracted them  simultaneously.  Dawdling  was  thus  eliminated,  and 
time  economized. 

83  No  such  comparison  has  been  made.  In  a  test  necessarily  oral 
on  account  of  the  main  objects  it  sought  to  accomplish  and  the  inability 
of  beginners  to  write  with  sufficient  ease,  it  was  impracticable  to  allow 
each  class  an  equal  period  of  time,  and  to  make  comparison  on  the  basis 
of  the  number  of  facts  written  or  of  examples  solved — the  only  mode 
of  procedure  by  which  can  be  obtained  the  uniform  results  necessary 
to  an  exact  comparison. 


BRA  FTp 
OF  THE 

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